id	sid	tid	token	lemma	pos
m-904	1	1	abstract	abstract	ADJ
m-904	1	2	ordinary	ordinary	ADJ
m-904	1	3	differential	differential	ADJ
m-904	1	4	equations	equation	NOUN
m-904	1	5	,	,	PUNCT
m-904	1	6	both	both	PRON
m-904	1	7	first	first	ADJ
m-904	1	8	and	and	CCONJ
m-904	1	9	second	second	ADJ
m-904	1	10	order	order	NOUN
m-904	1	11	,	,	PUNCT
m-904	1	12	are	be	AUX
m-904	1	13	essential	essential	ADJ
m-904	1	14	in	in	ADP
m-904	1	15	the	the	DET
m-904	1	16	modeling	modeling	NOUN
m-904	1	17	of	of	ADP
m-904	1	18	many	many	ADJ
m-904	1	19	physical	physical	ADJ
m-904	1	20	systems	system	NOUN
m-904	1	21	.	.	PUNCT
m-904	2	1	a	a	DET
m-904	2	2	system	system	NOUN
m-904	2	3	of	of	ADP
m-904	2	4	simultaneous	simultaneous	ADJ
m-904	2	5	differential	differential	ADJ
m-904	2	6	equations	equation	NOUN
m-904	2	7	results	result	VERB
m-904	2	8	from	from	ADP
m-904	2	9	more	more	ADV
m-904	2	10	complicated	complicated	ADJ
m-904	2	11	modeling	modeling	NOUN
m-904	2	12	involving	involve	VERB
m-904	2	13	more	more	ADJ
m-904	2	14	than	than	ADP
m-904	2	15	one	one	NUM
m-904	2	16	dependent	dependent	ADJ
m-904	2	17	variables	variable	NOUN
m-904	2	18	with	with	ADP
m-904	2	19	respect	respect	NOUN
m-904	2	20	to	to	ADP
m-904	2	21	a	a	DET
m-904	2	22	single	single	ADJ
m-904	2	23	independent	independent	ADJ
m-904	2	24	variable	variable	NOUN
m-904	2	25	.	.	PUNCT
m-904	3	1	there	there	PRON
m-904	3	2	are	be	VERB
m-904	3	3	several	several	ADJ
m-904	3	4	methods	method	NOUN
m-904	3	5	in	in	ADP
m-904	3	6	solving	solve	VERB
m-904	3	7	a	a	DET
m-904	3	8	system	system	NOUN
m-904	3	9	of	of	ADP
m-904	3	10	simultaneous	simultaneous	ADJ
m-904	3	11	linear	linear	ADJ
m-904	3	12	differential	differential	NOUN
m-904	3	13	equations	equation	NOUN
m-904	3	14	including	include	VERB
m-904	3	15	variable	variable	ADJ
m-904	3	16	substitution	substitution	NOUN
m-904	3	17	,	,	PUNCT
m-904	3	18	laplace	laplace	NOUN
m-904	3	19	transform	transform	NOUN
m-904	3	20	and	and	CCONJ
m-904	3	21	using	use	VERB
m-904	3	22	the	the	DET
m-904	3	23	d	d	NOUN
m-904	3	24	-	-	NOUN
m-904	3	25	operator	operator	NOUN
m-904	3	26	.	.	PUNCT
m-904	4	1	proposed	propose	VERB
m-904	4	2	in	in	ADP
m-904	4	3	this	this	DET
m-904	4	4	paper	paper	NOUN
m-904	4	5	is	be	AUX
m-904	4	6	a	a	DET
m-904	4	7	simplified	simplified	ADJ
m-904	4	8	method	method	NOUN
m-904	4	9	of	of	ADP
m-904	4	10	solving	solve	VERB
m-904	4	11	a	a	DET
m-904	4	12	set	set	NOUN
m-904	4	13	of	of	ADP
m-904	4	14	two	two	NUM
m-904	4	15	non	non	ADJ
m-904	4	16	-	-	ADJ
m-904	4	17	homogeneous	homogeneous	ADJ
m-904	4	18	linear	linear	ADJ
m-904	4	19	first	first	ADJ
m-904	4	20	-	-	PUNCT
m-904	4	21	order	order	NOUN
m-904	4	22	simultaneous	simultaneous	ADJ
m-904	4	23	ordinary	ordinary	ADJ
m-904	4	24	differential	differential	ADJ
m-904	4	25	equations	equation	NOUN
m-904	4	26	with	with	ADP
m-904	4	27	constant	constant	ADJ
m-904	4	28	coefficients	coefficient	NOUN
m-904	4	29	that	that	PRON
m-904	4	30	falls	fall	VERB
m-904	4	31	into	into	ADP
m-904	4	32	a	a	DET
m-904	4	33	certain	certain	ADJ
m-904	4	34	form	form	NOUN
m-904	4	35	.	.	PUNCT
m-904	5	1	an	an	DET
m-904	5	2	algebraic	algebraic	ADJ
m-904	5	3	formula	formula	NOUN
m-904	5	4	is	be	AUX
m-904	5	5	developed	develop	VERB
m-904	5	6	to	to	PART
m-904	5	7	compute	compute	VERB
m-904	5	8	the	the	DET
m-904	5	9	solution	solution	NOUN
m-904	5	10	to	to	ADP
m-904	5	11	the	the	DET
m-904	5	12	said	say	VERB
m-904	5	13	differential	differential	ADJ
m-904	5	14	equations	equation	NOUN
m-904	5	15	provided	provide	VERB
m-904	5	16	a	a	DET
m-904	5	17	certain	certain	ADJ
m-904	5	18	necessary	necessary	ADJ
m-904	5	19	condition	condition	NOUN
m-904	5	20	is	be	AUX
m-904	5	21	satisfied	satisfied	ADJ
m-904	5	22	.	.	PUNCT
m-904	6	1	four	four	NUM
m-904	6	2	different	different	ADJ
m-904	6	3	forms	form	NOUN
m-904	6	4	of	of	ADP
m-904	6	5	the	the	DET
m-904	6	6	functions	function	NOUN
m-904	6	7	of	of	ADP
m-904	6	8	the	the	DET
m-904	6	9	independent	independent	ADJ
m-904	6	10	variable	variable	NOUN
m-904	6	11	on	on	ADP
m-904	6	12	the	the	DET
m-904	6	13	right	right	ADJ
m-904	6	14	side	side	NOUN
m-904	6	15	of	of	ADP
m-904	6	16	the	the	DET
m-904	6	17	equations	equation	NOUN
m-904	6	18	,	,	PUNCT
m-904	6	19	namely	namely	ADV
m-904	6	20	constants	constant	NOUN
m-904	6	21	,	,	PUNCT
m-904	6	22	linear	linear	ADJ
m-904	6	23	functions	function	NOUN
m-904	6	24	,	,	PUNCT
m-904	6	25	natural	natural	ADJ
m-904	6	26	exponential	exponential	ADJ
m-904	6	27	functions	function	NOUN
m-904	6	28	and	and	CCONJ
m-904	6	29	sinusoidal	sinusoidal	NOUN
m-904	6	30	functions	function	NOUN
m-904	6	31	,	,	PUNCT
m-904	6	32	are	be	AUX
m-904	6	33	considered	consider	VERB
m-904	6	34	.	.	PUNCT
m-904	7	1	for	for	ADP
m-904	7	2	each	each	DET
m-904	7	3	case	case	NOUN
m-904	7	4	,	,	PUNCT
m-904	7	5	an	an	DET
m-904	7	6	algebraic	algebraic	ADJ
m-904	7	7	formula	formula	NOUN
m-904	7	8	to	to	PART
m-904	7	9	calculate	calculate	VERB
m-904	7	10	the	the	DET
m-904	7	11	dependent	dependent	ADJ
m-904	7	12	variable	variable	NOUN
m-904	7	13	as	as	ADV
m-904	7	14	well	well	ADV
m-904	7	15	as	as	ADP
m-904	7	16	its	its	PRON
m-904	7	17	derivative	derivative	NOUN
m-904	7	18	are	be	AUX
m-904	7	19	developed	develop	VERB
m-904	7	20	.	.	PUNCT
m-904	8	1	moreover	moreover	ADV
m-904	8	2	,	,	PUNCT
m-904	8	3	these	these	DET
m-904	8	4	simple	simple	ADJ
m-904	8	5	algebraic	algebraic	ADJ
m-904	8	6	formulae	formulae	NOUN
m-904	8	7	can	can	AUX
m-904	8	8	be	be	AUX
m-904	8	9	easily	easily	ADV
m-904	8	10	programmed	program	VERB
m-904	8	11	into	into	ADP
m-904	8	12	spreadsheet	spreadsheet	NOUN
m-904	8	13	where	where	SCONJ
m-904	8	14	one	one	NOUN
m-904	8	15	just	just	ADV
m-904	8	16	has	have	VERB
m-904	8	17	to	to	PART
m-904	8	18	enter	enter	VERB
m-904	8	19	the	the	DET
m-904	8	20	values	value	NOUN
m-904	8	21	of	of	ADP
m-904	8	22	the	the	DET
m-904	8	23	constants	constant	NOUN
m-904	8	24	and	and	CCONJ
m-904	8	25	coefficients	coefficient	NOUN
m-904	8	26	from	from	ADP
m-904	8	27	the	the	DET
m-904	8	28	original	original	ADJ
m-904	8	29	equations	equation	NOUN
m-904	8	30	and	and	CCONJ
m-904	8	31	instantly	instantly	ADV
m-904	8	32	obtain	obtain	VERB
m-904	8	33	the	the	DET
m-904	8	34	correct	correct	ADJ
m-904	8	35	answers	answer	NOUN
m-904	8	36	.	.	PUNCT
m-904	9	1	key	key	ADJ
m-904	9	2	words	word	NOUN
m-904	9	3	:	:	PUNCT
m-904	9	4	simultaneous	simultaneous	ADJ
m-904	9	5	differential	differential	NOUN
m-904	9	6	equations	equation	NOUN
m-904	9	7	;	;	PUNCT
m-904	9	8	non	non	ADJ
m-904	9	9	-	-	ADJ
m-904	9	10	homogeneous	homogeneous	ADJ
m-904	9	11	differential	differential	ADJ
m-904	9	12	equations	equation	NOUN
m-904	9	13	;	;	PUNCT
m-904	9	14	cramer	cramer	PROPN
m-904	9	15	’s	’s	PART
m-904	9	16	rule	rule	NOUN
m-904	9	17	.	.	PUNCT
m-904	10	1	ijo	ijo	PROPN
m-904	10	2	international	international	PROPN
m-904	10	3	journal	journal	PROPN
m-904	10	4	of	of	ADP
m-904	10	5	mathematics	mathematics	PROPN
m-904	10	6	(	(	PUNCT
m-904	10	7	issn	issn	PROPN
m-904	10	8	:	:	PUNCT
m-904	10	9	2992	2992	NUM
m-904	10	10	-	-	SYM
m-904	10	11	4421	4421	NUM
m-904	10	12	)	)	PUNCT
m-904	10	13	ijo	ijo	PROPN
m-904	10	14	journals	journal	NOUN
m-904	10	15	volume	volume	NOUN
m-904	10	16	07	07	NUM
m-904	10	17	|	|	NOUN
m-904	10	18	issue	issue	NOUN
m-904	10	19	07	07	NUM
m-904	11	1	|	|	CCONJ
m-904	11	2	july	july	PROPN
m-904	11	3	2024	2024	NUM
m-904	11	4	|	|	ADV
m-904	11	5	https://ijojournals.com/index.php/m/index	https://ijojournals.com/index.php/m/index	NOUN
m-904	11	6	1	1	NUM
m-904	11	7	algebraic	algebraic	ADJ
m-904	11	8	solution	solution	NOUN
m-904	11	9	to	to	ADP
m-904	11	10	simultaneous	simultaneous	ADJ
m-904	11	11	linear	linear	ADJ
m-904	11	12	first	first	ADJ
m-904	11	13	-	-	PUNCT
m-904	11	14	order	order	NOUN
m-904	11	15	nonhomogeneous	nonhomogeneous	ADJ
m-904	11	16	differential	differential	ADJ
m-904	11	17	equations	equation	NOUN
m-904	11	18	with	with	ADP
m-904	11	19	constant	constant	ADJ
m-904	11	20	coefficients	coefficient	NOUN
m-904	11	21	seng	seng	PROPN
m-904	11	22	chu	chu	PROPN
m-904	11	23	chow	chow	PROPN
m-904	11	24	inti	inti	PROPN
m-904	11	25	international	international	PROPN
m-904	11	26	college	college	PROPN
m-904	11	27	subang	subang	PROPN
m-904	11	28	,	,	PUNCT
m-904	11	29	malaysia	malaysia	PROPN
m-904	11	30	1	1	NUM
m-904	11	31	.	.	PUNCT
m-904	12	1	introduction	introduction	NOUN
m-904	12	2	ordinary	ordinary	ADJ
m-904	12	3	differential	differential	ADJ
m-904	12	4	equations	equation	NOUN
m-904	12	5	(	(	PUNCT
m-904	12	6	ode	ode	PROPN
m-904	12	7	)	)	PUNCT
m-904	12	8	play	play	VERB
m-904	12	9	a	a	DET
m-904	12	10	crucial	crucial	ADJ
m-904	12	11	role	role	NOUN
m-904	12	12	in	in	ADP
m-904	12	13	understanding	understanding	NOUN
m-904	12	14	and	and	CCONJ
m-904	12	15	modeling	model	VERB
m-904	12	16	various	various	ADJ
m-904	12	17	physical	physical	ADJ
m-904	12	18	phenomena	phenomenon	NOUN
m-904	12	19	across	across	ADP
m-904	12	20	many	many	ADJ
m-904	12	21	fields	field	NOUN
m-904	12	22	such	such	ADJ
m-904	12	23	as	as	ADP
m-904	12	24	engineering	engineering	NOUN
m-904	12	25	,	,	PUNCT
m-904	12	26	sciences	science	NOUN
m-904	12	27	and	and	CCONJ
m-904	12	28	economics	economic	NOUN
m-904	12	29	.	.	PUNCT
m-904	13	1	among	among	ADP
m-904	13	2	the	the	DET
m-904	13	3	numerous	numerous	ADJ
m-904	13	4	applications	application	NOUN
m-904	13	5	of	of	ADP
m-904	13	6	first	first	ADJ
m-904	13	7	-	-	PUNCT
m-904	13	8	order	order	NOUN
m-904	13	9	differential	differential	ADJ
m-904	13	10	equations	equation	NOUN
m-904	13	11	are	be	AUX
m-904	13	12	solutions	solution	NOUN
m-904	13	13	mixing	mix	VERB
m-904	13	14	,	,	PUNCT
m-904	13	15	population	population	NOUN
m-904	13	16	change	change	NOUN
m-904	13	17	,	,	PUNCT
m-904	13	18	heating	heating	NOUN
m-904	13	19	or	or	CCONJ
m-904	13	20	cooling	cooling	NOUN
m-904	13	21	,	,	PUNCT
m-904	13	22	free	free	ADJ
m-904	13	23	-	-	PUNCT
m-904	13	24	falling	fall	VERB
m-904	13	25	body	body	NOUN
m-904	13	26	,	,	PUNCT
m-904	13	27	fluid	fluid	NOUN
m-904	13	28	dynamics	dynamic	NOUN
m-904	13	29	,	,	PUNCT
m-904	13	30	resistive	resistive	NOUN
m-904	13	31	-	-	PUNCT
m-904	13	32	capacitive	capacitive	ADJ
m-904	13	33	(	(	PUNCT
m-904	13	34	rc	rc	PROPN
m-904	13	35	)	)	PUNCT
m-904	13	36	and	and	CCONJ
m-904	13	37	resistive	resistive	NOUN
m-904	13	38	-	-	PUNCT
m-904	13	39	inductive	inductive	ADJ
m-904	13	40	(	(	PUNCT
m-904	13	41	rl	rl	NOUN
m-904	13	42	)	)	PUNCT
m-904	13	43	circuits	circuit	NOUN
m-904	13	44	.	.	PUNCT
m-904	14	1	second	second	ADJ
m-904	14	2	-	-	PUNCT
m-904	14	3	order	order	NOUN
m-904	14	4	differential	differential	ADJ
m-904	14	5	equations	equation	NOUN
m-904	14	6	find	find	VERB
m-904	14	7	its	its	PRON
m-904	14	8	applications	application	NOUN
m-904	14	9	in	in	ADP
m-904	14	10	spring	spring	NOUN
m-904	14	11	-	-	PUNCT
m-904	14	12	mass	mass	NOUN
m-904	14	13	vibration	vibration	NOUN
m-904	14	14	,	,	PUNCT
m-904	14	15	rlc	rlc	PROPN
m-904	14	16	circuits	circuit	NOUN
m-904	14	17	,	,	PUNCT
m-904	14	18	wave	wave	NOUN
m-904	14	19	propagations	propagation	NOUN
m-904	14	20	,	,	PUNCT
m-904	14	21	etc	etc	X
m-904	14	22	.	.	X
m-904	14	23	by	by	ADP
m-904	14	24	solving	solve	VERB
m-904	14	25	a	a	DET
m-904	14	26	differential	differential	ADJ
m-904	14	27	equation	equation	NOUN
m-904	14	28	using	use	VERB
m-904	14	29	its	its	PRON
m-904	14	30	known	know	VERB
m-904	14	31	initial	initial	ADJ
m-904	14	32	conditions	condition	NOUN
m-904	14	33	,	,	PUNCT
m-904	14	34	which	which	PRON
m-904	14	35	is	be	AUX
m-904	14	36	known	know	VERB
m-904	14	37	as	as	ADP
m-904	14	38	initial	initial	ADJ
m-904	14	39	value	value	NOUN
m-904	14	40	problem	problem	NOUN
m-904	14	41	(	(	PUNCT
m-904	14	42	ivp	ivp	NOUN
m-904	14	43	)	)	PUNCT
m-904	14	44	,	,	PUNCT
m-904	14	45	we	we	PRON
m-904	14	46	can	can	AUX
m-904	14	47	anticipate	anticipate	VERB
m-904	14	48	the	the	DET
m-904	14	49	behavior	behavior	NOUN
m-904	14	50	of	of	ADP
m-904	14	51	the	the	DET
m-904	14	52	physical	physical	ADJ
m-904	14	53	system	system	NOUN
m-904	14	54	over	over	ADP
m-904	14	55	time	time	NOUN
m-904	14	56	.	.	PUNCT
m-904	15	1	[	[	X
m-904	15	2	7	7	X
m-904	15	3	]	]	X
m-904	15	4	simultaneous	simultaneous	ADJ
m-904	15	5	differential	differential	ADJ
m-904	15	6	equations	equation	NOUN
m-904	15	7	is	be	AUX
m-904	15	8	a	a	DET
m-904	15	9	system	system	NOUN
m-904	15	10	of	of	ADP
m-904	15	11	at	at	ADV
m-904	15	12	least	least	ADV
m-904	15	13	two	two	NUM
m-904	15	14	differential	differential	ADJ
m-904	15	15	equations	equation	NOUN
m-904	15	16	with	with	ADP
m-904	15	17	two	two	NUM
m-904	15	18	or	or	CCONJ
m-904	15	19	more	more	ADV
m-904	15	20	dependent	dependent	ADJ
m-904	15	21	variables	variable	NOUN
m-904	15	22	but	but	CCONJ
m-904	15	23	share	share	VERB
m-904	15	24	a	a	DET
m-904	15	25	single	single	ADJ
m-904	15	26	common	common	ADJ
m-904	15	27	independent	independent	ADJ
m-904	15	28	variable	variable	NOUN
m-904	15	29	.	.	PUNCT
m-904	16	1	these	these	DET
m-904	16	2	equations	equation	NOUN
m-904	16	3	govern	govern	VERB
m-904	16	4	the	the	DET
m-904	16	5	inter	inter	NOUN
m-904	16	6	-	-	NOUN
m-904	16	7	relationship	relationship	NOUN
m-904	16	8	between	between	ADP
m-904	16	9	the	the	DET
m-904	16	10	rate	rate	NOUN
m-904	16	11	of	of	ADP
m-904	16	12	change	change	NOUN
m-904	16	13	of	of	ADP
m-904	16	14	the	the	DET
m-904	16	15	dependent	dependent	ADJ
m-904	16	16	variables	variable	NOUN
m-904	16	17	with	with	ADP
m-904	16	18	respect	respect	NOUN
m-904	16	19	to	to	ADP
m-904	16	20	the	the	DET
m-904	16	21	independent	independent	ADJ
m-904	16	22	variable	variable	NOUN
m-904	16	23	.	.	PUNCT
m-904	17	1	[	[	X
m-904	17	2	8	8	NUM
m-904	17	3	]	]	PUNCT
m-904	17	4	the	the	DET
m-904	17	5	general	general	ADJ
m-904	17	6	form	form	NOUN
m-904	17	7	of	of	ADP
m-904	17	8	an	an	DET
m-904	17	9	nth	nth	NOUN
m-904	17	10	-	-	PUNCT
m-904	17	11	order	order	NOUN
m-904	17	12	linear	linear	ADJ
m-904	17	13	ordinary	ordinary	ADJ
m-904	17	14	differential	differential	ADJ
m-904	17	15	equation	equation	NOUN
m-904	17	16	is	be	AUX
m-904	17	17	:	:	PUNCT
m-904	17	18	𝑎𝑛(𝑥	𝑎𝑛(𝑥	X
m-904	17	19	)	)	PUNCT
m-904	17	20	𝑑𝑛𝑦	𝑑𝑛𝑦	NOUN
m-904	17	21	𝑑𝑥𝑛	𝑑𝑥𝑛	VERB
m-904	17	22	+	+	SYM
m-904	17	23	𝑎𝑛−1(𝑥	𝑎𝑛−1(𝑥	NOUN
m-904	17	24	)	)	PUNCT
m-904	17	25	𝑑𝑛−1𝑦	𝑑𝑛−1𝑦	NOUN
m-904	18	1	𝑑𝑥𝑛−1	𝑑𝑥𝑛−1	NOUN
m-904	18	2	+	+	SYM
m-904	18	3	⋯	⋯	PROPN
m-904	18	4	+	+	NOUN
m-904	18	5	𝑎1(𝑥	𝑎1(𝑥	NUM
m-904	18	6	)	)	PUNCT
m-904	18	7	𝑑𝑦	𝑑𝑦	ADP
m-904	18	8	𝑑𝑥	𝑑𝑥	ADP
m-904	18	9	+	+	NUM
m-904	18	10	𝑎0(𝑥)𝑦	𝑎0(𝑥)𝑦	NUM
m-904	18	11	=	=	SYM
m-904	18	12	𝐹(𝑥	𝐹(𝑥	NUM
m-904	18	13	)	)	PUNCT
m-904	18	14	(	(	PUNCT
m-904	18	15	1	1	X
m-904	18	16	)	)	PUNCT
m-904	18	17	since	since	SCONJ
m-904	18	18	this	this	DET
m-904	18	19	paper	paper	NOUN
m-904	18	20	only	only	ADV
m-904	18	21	consider	consider	VERB
m-904	18	22	1st	1st	ADJ
m-904	18	23	-	-	PUNCT
m-904	18	24	order	order	NOUN
m-904	18	25	ode	ode	ADJ
m-904	18	26	,	,	PUNCT
m-904	18	27	equation	equation	NOUN
m-904	18	28	(	(	PUNCT
m-904	18	29	1	1	X
m-904	18	30	)	)	PUNCT
m-904	18	31	reduces	reduce	VERB
m-904	18	32	to	to	ADP
m-904	18	33	𝑎1(𝑥	𝑎1(𝑥	NUM
m-904	18	34	)	)	PUNCT
m-904	18	35	𝑑𝑦	𝑑𝑦	ADP
m-904	18	36	𝑑𝑥	𝑑𝑥	ADP
m-904	18	37	+	+	NUM
m-904	18	38	𝑎0(𝑥)𝑦	𝑎0(𝑥)𝑦	NUM
m-904	18	39	=	=	SYM
m-904	18	40	𝐹(𝑥	𝐹(𝑥	NUM
m-904	18	41	)	)	PUNCT
m-904	18	42	(	(	PUNCT
m-904	18	43	2	2	X
m-904	18	44	)	)	PUNCT
m-904	18	45	where	where	SCONJ
m-904	18	46	𝑎1(𝑥	𝑎1(𝑥	NUM
m-904	18	47	)	)	PUNCT
m-904	18	48	,	,	PUNCT
m-904	18	49	𝑎2(𝑥	𝑎2(𝑥	NOUN
m-904	18	50	)	)	PUNCT
m-904	18	51	and	and	CCONJ
m-904	18	52	𝐹(𝑥	𝐹(𝑥	NUM
m-904	18	53	)	)	PUNCT
m-904	18	54	are	be	AUX
m-904	18	55	functions	function	NOUN
m-904	18	56	of	of	ADP
m-904	18	57	x	x	PUNCT
m-904	18	58	only	only	ADV
m-904	18	59	,	,	PUNCT
m-904	18	60	and	and	CCONJ
m-904	18	61	there	there	PRON
m-904	18	62	is	be	VERB
m-904	18	63	no	no	DET
m-904	18	64	restriction	restriction	NOUN
m-904	18	65	on	on	ADP
m-904	18	66	the	the	DET
m-904	18	67	nature	nature	NOUN
m-904	18	68	of	of	ADP
m-904	18	69	these	these	DET
m-904	18	70	x	x	NOUN
m-904	18	71	-	-	NOUN
m-904	18	72	dependencies	dependency	NOUN
m-904	18	73	.	.	PUNCT
m-904	19	1	[	[	X
m-904	19	2	5	5	NUM
m-904	19	3	,	,	PUNCT
m-904	19	4	6	6	NUM
m-904	19	5	]	]	X
m-904	19	6	ijo	ijo	PROPN
m-904	19	7	international	international	PROPN
m-904	19	8	journal	journal	PROPN
m-904	19	9	of	of	ADP
m-904	19	10	mathematics	mathematics	PROPN
m-904	19	11	(	(	PUNCT
m-904	19	12	issn	issn	PROPN
m-904	19	13	:	:	PUNCT
m-904	19	14	2992	2992	NUM
m-904	19	15	-	-	SYM
m-904	19	16	4421	4421	NUM
m-904	19	17	)	)	PUNCT
m-904	19	18	ijo	ijo	PROPN
m-904	19	19	journals	journal	NOUN
m-904	19	20	volume	volume	NOUN
m-904	19	21	07	07	NUM
m-904	19	22	|	|	NOUN
m-904	19	23	issue	issue	NOUN
m-904	19	24	07	07	NUM
m-904	20	1	|	|	CCONJ
m-904	20	2	july	july	PROPN
m-904	20	3	2024	2024	NUM
m-904	20	4	|	|	ADV
m-904	20	5	https://ijojournals.com/index.php/m/index	https://ijojournals.com/index.php/m/index	NOUN
m-904	20	6	2	2	NUM
m-904	20	7	2	2	NUM
m-904	20	8	example	example	NOUN
m-904	20	9	of	of	ADP
m-904	20	10	a	a	DET
m-904	20	11	system	system	NOUN
m-904	20	12	of	of	ADP
m-904	20	13	simultaneous	simultaneous	ADJ
m-904	20	14	1st	1st	ADJ
m-904	20	15	-	-	PUNCT
m-904	20	16	order	order	NOUN
m-904	20	17	ode	ode	ADJ
m-904	20	18	2.1	2.1	NUM
m-904	20	19	double	double	ADJ
m-904	20	20	mixing	mixing	NOUN
m-904	20	21	problem	problem	NOUN
m-904	20	22	figure	figure	NOUN
m-904	20	23	1	1	NUM
m-904	20	24	below	below	ADV
m-904	20	25	shows	show	VERB
m-904	20	26	a	a	DET
m-904	20	27	brine	brine	NOUN
m-904	20	28	solution	solution	NOUN
m-904	20	29	with	with	ADP
m-904	20	30	concentration	concentration	NOUN
m-904	20	31	𝐵1(𝑘𝑔/𝑙	𝐵1(𝑘𝑔/𝑙	NOUN
m-904	20	32	)	)	PUNCT
m-904	20	33	flows	flow	VERB
m-904	20	34	at	at	ADP
m-904	20	35	a	a	DET
m-904	20	36	constant	constant	ADJ
m-904	20	37	rate	rate	NOUN
m-904	20	38	of	of	ADP
m-904	20	39	𝐹1(𝑙/𝑠	𝐹1(𝑙/𝑠	NUM
m-904	20	40	)	)	PUNCT
m-904	20	41	into	into	ADP
m-904	20	42	tank	tank	NOUN
m-904	20	43	1	1	NUM
m-904	20	44	that	that	PRON
m-904	20	45	initially	initially	ADV
m-904	20	46	contains	contain	VERB
m-904	20	47	𝑉1(𝑙	𝑉1(𝑙	PROPN
m-904	20	48	)	)	PUNCT
m-904	20	49	of	of	ADP
m-904	20	50	the	the	DET
m-904	20	51	same	same	ADJ
m-904	20	52	brine	brine	NOUN
m-904	20	53	.	.	PUNCT
m-904	21	1	the	the	DET
m-904	21	2	solution	solution	NOUN
m-904	21	3	in	in	ADP
m-904	21	4	the	the	DET
m-904	21	5	tank	tank	NOUN
m-904	21	6	is	be	AUX
m-904	21	7	kept	keep	VERB
m-904	21	8	well	well	ADV
m-904	21	9	stirred	stir	VERB
m-904	21	10	and	and	CCONJ
m-904	21	11	flows	flow	VERB
m-904	21	12	out	out	ADP
m-904	21	13	of	of	ADP
m-904	21	14	tank	tank	NOUN
m-904	21	15	1	1	NUM
m-904	21	16	at	at	ADP
m-904	21	17	a	a	DET
m-904	21	18	rate	rate	NOUN
m-904	21	19	of	of	ADP
m-904	21	20	𝐹2	𝐹2	NOUN
m-904	21	21	with	with	ADP
m-904	21	22	concentration	concentration	NOUN
m-904	21	23	𝐵2	𝐵2	NOUN
m-904	21	24	directly	directly	ADV
m-904	21	25	into	into	ADP
m-904	21	26	tank	tank	NOUN
m-904	21	27	2	2	NUM
m-904	21	28	with	with	ADP
m-904	21	29	volume	volume	NOUN
m-904	21	30	𝑉2	𝑉2	NOUN
m-904	21	31	.	.	PUNCT
m-904	22	1	the	the	DET
m-904	22	2	solution	solution	NOUN
m-904	22	3	in	in	ADP
m-904	22	4	tank	tank	NOUN
m-904	22	5	2	2	NUM
m-904	22	6	is	be	AUX
m-904	22	7	also	also	ADV
m-904	22	8	well	well	ADV
m-904	22	9	mixed	mixed	ADJ
m-904	22	10	and	and	CCONJ
m-904	22	11	flows	flow	VERB
m-904	22	12	out	out	ADP
m-904	22	13	at	at	ADP
m-904	22	14	a	a	DET
m-904	22	15	rate	rate	NOUN
m-904	22	16	of	of	ADP
m-904	22	17	𝐹3	𝐹3	PROPN
m-904	22	18	with	with	ADP
m-904	22	19	concentration	concentration	NOUN
m-904	22	20	𝐵3	𝐵3	NOUN
m-904	22	21	.	.	PUNCT
m-904	23	1	the	the	DET
m-904	23	2	mass	mass	NOUN
m-904	23	3	of	of	ADP
m-904	23	4	salt	salt	NOUN
m-904	23	5	in	in	ADP
m-904	23	6	tank	tank	NOUN
m-904	23	7	1	1	NUM
m-904	23	8	and	and	CCONJ
m-904	23	9	tank	tank	NOUN
m-904	23	10	2	2	NUM
m-904	23	11	are	be	AUX
m-904	23	12	𝑚1(𝑡	𝑚1(𝑡	PRON
m-904	23	13	)	)	PUNCT
m-904	23	14	and	and	CCONJ
m-904	23	15	𝑚2(𝑡	𝑚2(𝑡	NOUN
m-904	23	16	)	)	PUNCT
m-904	23	17	respectively	respectively	ADV
m-904	23	18	.	.	PUNCT
m-904	24	1	the	the	DET
m-904	24	2	ivp	ivp	NOUN
m-904	24	3	for	for	ADP
m-904	24	4	this	this	DET
m-904	24	5	problem	problem	NOUN
m-904	24	6	is	be	AUX
m-904	24	7	given	give	VERB
m-904	24	8	by	by	ADP
m-904	24	9	:	:	PUNCT
m-904	24	10	𝑑𝑚1	𝑑𝑚1	X
m-904	24	11	𝑑𝑡	𝑑𝑡	ADP
m-904	24	12	=	=	PUNCT
m-904	24	13	𝐵1𝐹1	𝐵1𝐹1	X
m-904	24	14	−	−	PROPN
m-904	24	15	𝑚1	𝑚1	NOUN
m-904	24	16	𝑉1+(𝐹1−𝐹2)∙𝑡	𝑉1+(𝐹1−𝐹2)∙𝑡	VERB
m-904	24	17	𝐹2	𝐹2	NOUN
m-904	24	18	,	,	PUNCT
m-904	24	19	𝑚1(0	𝑚1(0	PROPN
m-904	24	20	)	)	PUNCT
m-904	24	21	=	=	SYM
m-904	25	1	𝑎	𝑎	X
m-904	25	2	(	(	PUNCT
m-904	25	3	3a	3a	NUM
m-904	25	4	)	)	PUNCT
m-904	25	5	𝑑𝑚2	𝑑𝑚2	NOUN
m-904	25	6	𝑑𝑡	𝑑𝑡	ADP
m-904	25	7	=	=	PROPN
m-904	25	8	𝑚1	𝑚1	X
m-904	25	9	𝑉1+(𝐹1−𝐹2)∙𝑡	𝑉1+(𝐹1−𝐹2)∙𝑡	PUNCT
m-904	25	10	𝐹2	𝐹2	NOUN
m-904	25	11	−	−	PROPN
m-904	25	12	𝑚2	𝑚2	NOUN
m-904	25	13	𝑉2+(𝐹2−𝐹3)∙𝑡	𝑉2+(𝐹2−𝐹3)∙𝑡	NUM
m-904	25	14	𝐹3	𝐹3	NOUN
m-904	25	15	,	,	PUNCT
m-904	25	16	𝑚2(0	𝑚2(0	ADV
m-904	25	17	)	)	PUNCT
m-904	25	18	=	=	SYM
m-904	25	19	𝑏	𝑏	PROPN
m-904	25	20	(	(	PUNCT
m-904	25	21	3b	3b	NUM
m-904	25	22	)	)	PUNCT
m-904	25	23	where	where	SCONJ
m-904	25	24	a	a	PRON
m-904	25	25	and	and	CCONJ
m-904	25	26	b	b	NOUN
m-904	25	27	are	be	AUX
m-904	25	28	the	the	DET
m-904	25	29	initial	initial	ADJ
m-904	25	30	mass	mass	NOUN
m-904	25	31	of	of	ADP
m-904	25	32	salt	salt	NOUN
m-904	25	33	in	in	ADP
m-904	25	34	tank	tank	NOUN
m-904	25	35	1	1	NUM
m-904	25	36	and	and	CCONJ
m-904	25	37	tank	tank	NOUN
m-904	25	38	2	2	NUM
m-904	25	39	,	,	PUNCT
m-904	25	40	respectively	respectively	ADV
m-904	25	41	.	.	PUNCT
m-904	26	1	2.2	2.2	NUM
m-904	26	2	simplifying	simplify	VERB
m-904	26	3	ivp	ivp	NOUN
m-904	26	4	in	in	ADP
m-904	26	5	most	most	ADJ
m-904	26	6	applications	application	NOUN
m-904	26	7	,	,	PUNCT
m-904	26	8	𝐹1	𝐹1	PROPN
m-904	26	9	,	,	PUNCT
m-904	26	10	𝐹2	𝐹2	NOUN
m-904	26	11	and	and	CCONJ
m-904	26	12	𝐹3	𝐹3	PROPN
m-904	26	13	are	be	AUX
m-904	26	14	constant	constant	ADJ
m-904	26	15	and	and	CCONJ
m-904	26	16	equal	equal	ADJ
m-904	26	17	,	,	PUNCT
m-904	26	18	otherwise	otherwise	ADV
m-904	26	19	the	the	DET
m-904	26	20	volume	volume	NOUN
m-904	26	21	of	of	ADP
m-904	26	22	liquid	liquid	NOUN
m-904	26	23	in	in	ADP
m-904	26	24	the	the	DET
m-904	26	25	two	two	NUM
m-904	26	26	tanks	tank	NOUN
m-904	26	27	would	would	AUX
m-904	26	28	be	be	AUX
m-904	26	29	changing	change	VERB
m-904	26	30	with	with	ADP
m-904	26	31	time	time	NOUN
m-904	26	32	.	.	PUNCT
m-904	27	1	if	if	SCONJ
m-904	27	2	the	the	DET
m-904	27	3	two	two	NUM
m-904	27	4	volumes	volume	NOUN
m-904	27	5	are	be	AUX
m-904	27	6	assumed	assume	VERB
m-904	27	7	to	to	PART
m-904	27	8	be	be	AUX
m-904	27	9	equal	equal	ADJ
m-904	27	10	,	,	PUNCT
m-904	27	11	that	that	PRON
m-904	27	12	is	be	AUX
m-904	27	13	𝑉1	𝑉1	NOUN
m-904	27	14	=	=	SYM
m-904	27	15	𝑉2	𝑉2	NOUN
m-904	27	16	=	=	SYM
m-904	27	17	𝑉	𝑉	PROPN
m-904	27	18	,	,	PUNCT
m-904	27	19	then	then	ADV
m-904	27	20	the	the	DET
m-904	27	21	ivp	ivp	NOUN
m-904	27	22	in	in	ADP
m-904	27	23	equation	equation	NOUN
m-904	27	24	(	(	PUNCT
m-904	27	25	3	3	X
m-904	27	26	)	)	PUNCT
m-904	27	27	is	be	AUX
m-904	27	28	simplified	simplify	VERB
m-904	27	29	to	to	PART
m-904	27	30	𝑚1	𝑚1	VERB
m-904	27	31	′	′	NOUN
m-904	27	32	=	=	PUNCT
m-904	27	33	𝐴	𝐴	PROPN
m-904	27	34	−	−	PROPN
m-904	27	35	𝐾𝑚1	𝐾𝑚1	NOUN
m-904	27	36	(	(	PUNCT
m-904	27	37	4a	4a	NUM
m-904	27	38	)	)	PUNCT
m-904	27	39	𝑚2	𝑚2	NOUN
m-904	27	40	′	′	NUM
m-904	27	41	=	=	PUNCT
m-904	27	42	𝐾𝑚1	𝐾𝑚1	PART
m-904	28	1	−	−	PROPN
m-904	28	2	𝐾𝑚2	𝐾𝑚2	NOUN
m-904	28	3	(	(	PUNCT
m-904	28	4	4b	4b	PROPN
m-904	28	5	)	)	PUNCT
m-904	28	6	where	where	SCONJ
m-904	28	7	figure	figure	NOUN
m-904	28	8	1	1	NUM
m-904	28	9	:	:	PUNCT
m-904	28	10	double	double	ADJ
m-904	28	11	mixing	mixing	NOUN
m-904	28	12	problem	problem	NOUN
m-904	28	13	ijo	ijo	PROPN
m-904	28	14	international	international	PROPN
m-904	28	15	journal	journal	PROPN
m-904	28	16	of	of	ADP
m-904	28	17	mathematics	mathematics	PROPN
m-904	28	18	(	(	PUNCT
m-904	28	19	issn	issn	PROPN
m-904	28	20	:	:	PUNCT
m-904	28	21	2992	2992	NUM
m-904	28	22	-	-	SYM
m-904	28	23	4421	4421	NUM
m-904	28	24	)	)	PUNCT
m-904	28	25	ijo	ijo	PROPN
m-904	28	26	journals	journal	NOUN
m-904	28	27	volume	volume	NOUN
m-904	28	28	07	07	NUM
m-904	28	29	|	|	NOUN
m-904	28	30	issue	issue	NOUN
m-904	28	31	07	07	NUM
m-904	28	32	|	|	CCONJ
m-904	28	33	july	july	PROPN
m-904	28	34	2024	2024	NUM
m-904	29	1	|	|	ADV
m-904	29	2	https://ijojournals.com/index.php/m/index	https://ijojournals.com/index.php/m/index	NOUN
m-904	29	3	3	3	NUM
m-904	29	4	𝐴	𝐴	PROPN
m-904	29	5	=	=	SYM
m-904	29	6	𝐵1𝐹	𝐵1𝐹	PROPN
m-904	29	7	,	,	PUNCT
m-904	29	8	𝐾	𝐾	PROPN
m-904	29	9	=	=	SYM
m-904	29	10	𝐹	𝐹	PROPN
m-904	29	11	𝑉	𝑉	PROPN
m-904	29	12	and	and	CCONJ
m-904	29	13	𝐹1	𝐹1	NOUN
m-904	29	14	=	=	X
m-904	29	15	𝐹2	𝐹2	PROPN
m-904	29	16	=	=	PUNCT
m-904	29	17	𝐹3	𝐹3	PROPN
m-904	29	18	=	=	PUNCT
m-904	29	19	𝐹.	𝐹.	PROPN
m-904	29	20	(	(	PUNCT
m-904	29	21	4c	4c	NOUN
m-904	29	22	)	)	PUNCT
m-904	29	23	putting	put	VERB
m-904	29	24	equation	equation	NOUN
m-904	29	25	(	(	PUNCT
m-904	29	26	4	4	NUM
m-904	29	27	)	)	PUNCT
m-904	29	28	in	in	ADP
m-904	29	29	matrix	matrix	NOUN
m-904	29	30	form	form	NOUN
m-904	29	31	,	,	PUNCT
m-904	29	32	we	we	PRON
m-904	29	33	have	have	VERB
m-904	29	34	[	[	PUNCT
m-904	29	35	𝑚1	𝑚1	NOUN
m-904	29	36	′	′	NUM
m-904	29	37	𝑚2	𝑚2	PROPN
m-904	29	38	′	′	NUM
m-904	29	39	]	]	PUNCT
m-904	30	1	=	=	PUNCT
m-904	30	2	[	[	PUNCT
m-904	30	3	−𝐾	−𝐾	NOUN
m-904	30	4	0	0	NUM
m-904	30	5	𝐾	𝐾	PROPN
m-904	30	6	−𝐾	−𝐾	PROPN
m-904	30	7	]	]	PUNCT
m-904	30	8	[	[	PUNCT
m-904	30	9	𝑚1	𝑚1	NOUN
m-904	30	10	𝑚2	𝑚2	NOUN
m-904	30	11	]	]	PUNCT
m-904	31	1	+	+	CCONJ
m-904	31	2	[	[	PUNCT
m-904	31	3	𝐴	𝐴	NOUN
m-904	31	4	0	0	NUM
m-904	31	5	]	]	PUNCT
m-904	31	6	(	(	PUNCT
m-904	31	7	5	5	X
m-904	31	8	)	)	PUNCT
m-904	31	9	if	if	SCONJ
m-904	31	10	the	the	DET
m-904	31	11	input	input	NOUN
m-904	31	12	concentration	concentration	NOUN
m-904	31	13	can	can	AUX
m-904	31	14	be	be	AUX
m-904	31	15	continuously	continuously	ADV
m-904	31	16	adjusted	adjust	VERB
m-904	31	17	to	to	PART
m-904	31	18	be	be	AUX
m-904	31	19	in	in	ADP
m-904	31	20	direct	direct	ADJ
m-904	31	21	proportional	proportional	NOUN
m-904	31	22	to	to	ADP
m-904	31	23	the	the	DET
m-904	31	24	varying	vary	VERB
m-904	31	25	mass	mass	NOUN
m-904	31	26	of	of	ADP
m-904	31	27	salt	salt	NOUN
m-904	31	28	in	in	ADP
m-904	31	29	tank	tank	NOUN
m-904	31	30	1	1	NUM
m-904	31	31	,	,	PUNCT
m-904	31	32	that	that	PRON
m-904	31	33	is	be	AUX
m-904	31	34	𝐵1	𝐵1	PROPN
m-904	31	35	∝	∝	PROPN
m-904	31	36	𝑚1	𝑚1	PROPN
m-904	31	37	,	,	PUNCT
m-904	31	38	then	then	ADV
m-904	31	39	𝐴	𝐴	PROPN
m-904	31	40	=	=	SYM
m-904	31	41	𝑝𝑚1𝐹	𝑝𝑚1𝐹	PROPN
m-904	31	42	=	=	SYM
m-904	31	43	𝐶	𝐶	PROPN
m-904	31	44	∙	∙	PROPN
m-904	31	45	𝑚1	𝑚1	NOUN
m-904	31	46	(	(	PUNCT
m-904	31	47	6	6	NUM
m-904	31	48	)	)	PUNCT
m-904	31	49	where	where	SCONJ
m-904	31	50	p	p	NOUN
m-904	31	51	is	be	AUX
m-904	31	52	the	the	DET
m-904	31	53	constant	constant	ADJ
m-904	31	54	of	of	ADP
m-904	31	55	proportionality	proportionality	NOUN
m-904	31	56	between	between	ADP
m-904	31	57	𝐵1	𝐵1	NOUN
m-904	31	58	and	and	CCONJ
m-904	31	59	𝑚1	𝑚1	NOUN
m-904	31	60	,	,	PUNCT
m-904	31	61	and	and	CCONJ
m-904	31	62	the	the	DET
m-904	31	63	constant	constant	ADJ
m-904	31	64	c	c	NOUN
m-904	31	65	is	be	AUX
m-904	31	66	the	the	DET
m-904	31	67	product	product	NOUN
m-904	31	68	of	of	ADP
m-904	31	69	p	p	PROPN
m-904	31	70	and	and	CCONJ
m-904	31	71	f.	f.	PROPN
m-904	31	72	with	with	ADP
m-904	31	73	this	this	DET
m-904	31	74	assumption	assumption	NOUN
m-904	31	75	,	,	PUNCT
m-904	31	76	equation	equation	NOUN
m-904	31	77	(	(	PUNCT
m-904	31	78	5	5	X
m-904	31	79	)	)	PUNCT
m-904	31	80	can	can	AUX
m-904	31	81	be	be	AUX
m-904	31	82	simplified	simplify	VERB
m-904	31	83	to	to	ADP
m-904	31	84	[	[	PUNCT
m-904	31	85	𝑚1	𝑚1	NOUN
m-904	31	86	′	′	NUM
m-904	31	87	𝑚2	𝑚2	PROPN
m-904	31	88	′	′	NUM
m-904	31	89	]	]	PUNCT
m-904	32	1	=	=	PUNCT
m-904	32	2	[	[	PUNCT
m-904	32	3	−𝐾	−𝐾	PROPN
m-904	32	4	+	+	CCONJ
m-904	32	5	𝐶	𝐶	PROPN
m-904	32	6	0	0	PUNCT
m-904	32	7	𝐾	𝐾	PROPN
m-904	32	8	−𝐾	−𝐾	PROPN
m-904	32	9	]	]	PUNCT
m-904	32	10	[	[	PUNCT
m-904	32	11	𝑚1	𝑚1	NOUN
m-904	32	12	𝑚2	𝑚2	PROPN
m-904	32	13	]	]	PUNCT
m-904	32	14	(	(	PUNCT
m-904	32	15	7	7	X
m-904	32	16	)	)	PUNCT
m-904	32	17	2.3	2.3	NUM
m-904	32	18	de	de	NOUN
m-904	32	19	-	-	NOUN
m-904	32	20	coupling	coupling	NOUN
m-904	32	21	of	of	ADP
m-904	32	22	the	the	DET
m-904	32	23	system	system	NOUN
m-904	32	24	the	the	DET
m-904	32	25	eigenvalues	eigenvalue	NOUN
m-904	32	26	of	of	ADP
m-904	32	27	equation	equation	NOUN
m-904	32	28	(	(	PUNCT
m-904	32	29	7	7	X
m-904	32	30	)	)	PUNCT
m-904	32	31	can	can	AUX
m-904	32	32	be	be	AUX
m-904	32	33	found	find	VERB
m-904	32	34	by	by	ADP
m-904	32	35	solving	solve	VERB
m-904	32	36	the	the	DET
m-904	32	37	characteristic	characteristic	ADJ
m-904	32	38	polynomial	polynomial	NOUN
m-904	33	1	|	|	ADV
m-904	33	2	𝜆	𝜆	ADP
m-904	33	3	+	+	CCONJ
m-904	33	4	𝐾	𝐾	PROPN
m-904	33	5	−	−	PROPN
m-904	33	6	𝐶	𝐶	PROPN
m-904	33	7	0	0	PUNCT
m-904	33	8	−𝐾	−𝐾	PROPN
m-904	33	9	𝜆	𝜆	PROPN
m-904	33	10	+	+	PUNCT
m-904	33	11	𝐾	𝐾	NOUN
m-904	33	12	|	|	NOUN
m-904	33	13	=	=	SYM
m-904	33	14	0	0	PUNCT
m-904	34	1	(	(	PUNCT
m-904	34	2	8)	8)	NUM
m-904	34	3	which	which	PRON
m-904	34	4	yields	yield	VERB
m-904	34	5	𝜆1	𝜆1	NOUN
m-904	35	1	=	=	PUNCT
m-904	35	2	−𝐾	−𝐾	PROPN
m-904	35	3	and	and	CCONJ
m-904	35	4	𝜆2	𝜆2	NOUN
m-904	35	5	=	=	PUNCT
m-904	36	1	−(𝐾	−(𝐾	ADJ
m-904	36	2	−	−	PROPN
m-904	36	3	𝐶	𝐶	PROPN
m-904	36	4	)	)	PUNCT
m-904	36	5	(	(	PUNCT
m-904	36	6	9	9	NUM
m-904	36	7	)	)	PUNCT
m-904	36	8	with	with	ADP
m-904	36	9	the	the	DET
m-904	36	10	corresponding	correspond	VERB
m-904	36	11	eigenvectors	eigenvector	NOUN
m-904	36	12	𝛼1	𝛼1	NOUN
m-904	36	13	=	=	SYM
m-904	36	14	𝑟	𝑟	X
m-904	36	15	[	[	PUNCT
m-904	36	16	0	0	NUM
m-904	36	17	1	1	NUM
m-904	36	18	]	]	PUNCT
m-904	36	19	and	and	CCONJ
m-904	36	20	𝛼2	𝛼2	PROPN
m-904	36	21	=	=	SYM
m-904	36	22	𝑠	𝑠	PROPN
m-904	36	23	[	[	PUNCT
m-904	36	24	𝐶	𝐶	PROPN
m-904	36	25	𝐾	𝐾	PROPN
m-904	36	26	1	1	NUM
m-904	36	27	]	]	PUNCT
m-904	36	28	,	,	PUNCT
m-904	36	29	𝑟	𝑟	AUX
m-904	36	30	,	,	PUNCT
m-904	36	31	𝑠	𝑠	PROPN
m-904	36	32	∈	∈	PROPN
m-904	36	33	𝑅.	𝑅.	NOUN
m-904	36	34	(	(	PUNCT
m-904	36	35	10	10	NUM
m-904	36	36	)	)	PUNCT
m-904	36	37	the	the	DET
m-904	36	38	transition	transition	NOUN
m-904	36	39	matrix	matrix	NOUN
m-904	36	40	p	p	NOUN
m-904	36	41	and	and	CCONJ
m-904	36	42	its	its	PRON
m-904	36	43	inverse	inverse	NOUN
m-904	36	44	is	be	AUX
m-904	36	45	given	give	VERB
m-904	36	46	by	by	ADP
m-904	36	47	𝑃	𝑃	NOUN
m-904	36	48	=	=	SYM
m-904	36	49	[	[	PUNCT
m-904	36	50	0	0	NUM
m-904	36	51	𝐶	𝐶	PROPN
m-904	36	52	𝐾	𝐾	PROPN
m-904	36	53	1	1	NUM
m-904	36	54	1	1	NUM
m-904	36	55	]	]	PUNCT
m-904	36	56	,	,	PUNCT
m-904	36	57	𝑃−1	𝑃−1	X
m-904	36	58	=	=	SYM
m-904	37	1	1	1	NUM
m-904	37	2	−	−	PROPN
m-904	37	3	𝐶	𝐶	PROPN
m-904	37	4	𝐾	𝐾	PROPN
m-904	37	5	[	[	PUNCT
m-904	37	6	1	1	NUM
m-904	37	7	−	−	PROPN
m-904	37	8	𝐶	𝐶	PROPN
m-904	37	9	𝐾	𝐾	PROPN
m-904	37	10	−1	−1	NOUN
m-904	37	11	0	0	NUM
m-904	37	12	]	]	PUNCT
m-904	37	13	(	(	PUNCT
m-904	37	14	11	11	NUM
m-904	37	15	)	)	PUNCT
m-904	37	16	as	as	SCONJ
m-904	37	17	expected	expect	VERB
m-904	37	18	,	,	PUNCT
m-904	37	19	ijo	ijo	PROPN
m-904	37	20	international	international	PROPN
m-904	37	21	journal	journal	PROPN
m-904	37	22	of	of	ADP
m-904	37	23	mathematics	mathematics	PROPN
m-904	37	24	(	(	PUNCT
m-904	37	25	issn	issn	PROPN
m-904	37	26	:	:	PUNCT
m-904	37	27	2992	2992	NUM
m-904	37	28	-	-	SYM
m-904	37	29	4421	4421	NUM
m-904	37	30	)	)	PUNCT
m-904	37	31	ijo	ijo	PROPN
m-904	37	32	journals	journal	NOUN
m-904	37	33	volume	volume	NOUN
m-904	37	34	07	07	NUM
m-904	37	35	|	|	NOUN
m-904	37	36	issue	issue	NOUN
m-904	37	37	07	07	NUM
m-904	38	1	|	|	CCONJ
m-904	38	2	july	july	PROPN
m-904	38	3	2024	2024	NUM
m-904	38	4	|	|	ADV
m-904	38	5	https://ijojournals.com/index.php/m/index	https://ijojournals.com/index.php/m/index	NOUN
m-904	38	6	4	4	NUM
m-904	38	7	1	1	NUM
m-904	38	8	−	−	PROPN
m-904	38	9	𝐶	𝐶	PROPN
m-904	38	10	𝐾	𝐾	PROPN
m-904	38	11	[	[	PUNCT
m-904	38	12	1	1	NUM
m-904	38	13	−	−	PROPN
m-904	38	14	𝐶	𝐶	PROPN
m-904	38	15	𝐾	𝐾	PROPN
m-904	38	16	−1	−1	NOUN
m-904	38	17	0	0	PUNCT
m-904	38	18	]	]	PUNCT
m-904	39	1	[	[	PUNCT
m-904	39	2	−𝐾	−𝐾	PROPN
m-904	39	3	+	+	CCONJ
m-904	39	4	𝐶	𝐶	PROPN
m-904	39	5	0	0	PUNCT
m-904	39	6	𝐾	𝐾	PROPN
m-904	39	7	−𝐾	−𝐾	VERB
m-904	39	8	]	]	PUNCT
m-904	40	1	[	[	PUNCT
m-904	40	2	0	0	NUM
m-904	40	3	𝐶	𝐶	PROPN
m-904	40	4	𝐾	𝐾	PROPN
m-904	40	5	1	1	NUM
m-904	40	6	1	1	NUM
m-904	40	7	]	]	PUNCT
m-904	40	8	=	=	PUNCT
m-904	40	9	[	[	PUNCT
m-904	40	10	𝜆1	𝜆1	NOUN
m-904	40	11	0	0	NUM
m-904	40	12	0	0	NUM
m-904	40	13	𝜆2	𝜆2	NOUN
m-904	40	14	]	]	PUNCT
m-904	40	15	(	(	PUNCT
m-904	40	16	12	12	NUM
m-904	40	17	)	)	PUNCT
m-904	40	18	let	let	VERB
m-904	40	19	’s	’s	PRON
m-904	40	20	define	define	VERB
m-904	40	21	𝑚	𝑚	NOUN
m-904	40	22	=	=	PUNCT
m-904	40	23	𝑃𝑥	𝑃𝑥	PROPN
m-904	40	24	and	and	CCONJ
m-904	40	25	𝑚′	𝑚′	NUM
m-904	40	26	=	=	SYM
m-904	40	27	𝑃𝑥′	𝑃𝑥′	PROPN
m-904	40	28	(	(	PUNCT
m-904	40	29	13	13	NUM
m-904	40	30	)	)	PUNCT
m-904	40	31	where	where	SCONJ
m-904	40	32	𝑥	𝑥	NOUN
m-904	40	33	=	=	X
m-904	40	34	[	[	PUNCT
m-904	40	35	𝑥1(𝑡	𝑥1(𝑡	NOUN
m-904	40	36	)	)	PUNCT
m-904	40	37	𝑥2(𝑡	𝑥2(𝑡	NOUN
m-904	40	38	)	)	PUNCT
m-904	40	39	]	]	PUNCT
m-904	40	40	is	be	AUX
m-904	40	41	a	a	DET
m-904	40	42	dummy	dummy	ADJ
m-904	40	43	function	function	NOUN
m-904	40	44	.	.	PUNCT
m-904	41	1	substitute	substitute	NOUN
m-904	41	2	(	(	PUNCT
m-904	41	3	13	13	NUM
m-904	41	4	)	)	PUNCT
m-904	41	5	into	into	ADP
m-904	41	6	(	(	PUNCT
m-904	41	7	7	7	X
m-904	41	8	)	)	PUNCT
m-904	41	9	yields	yield	NOUN
m-904	41	10	𝑃𝑥′	𝑃𝑥′	NOUN
m-904	41	11	=	=	PUNCT
m-904	42	1	[	[	PUNCT
m-904	42	2	−𝐾	−𝐾	PROPN
m-904	42	3	+	+	CCONJ
m-904	42	4	𝐶	𝐶	PROPN
m-904	42	5	0	0	PUNCT
m-904	42	6	𝐾	𝐾	PROPN
m-904	42	7	−𝐾	−𝐾	VERB
m-904	42	8	]	]	PUNCT
m-904	43	1	𝑃𝑥	𝑃𝑥	PROPN
m-904	43	2	(	(	PUNCT
m-904	43	3	14	14	NUM
m-904	43	4	)	)	PUNCT
m-904	43	5	multiply	multiply	ADV
m-904	43	6	(	(	PUNCT
m-904	43	7	14	14	NUM
m-904	43	8	)	)	PUNCT
m-904	43	9	by	by	ADP
m-904	43	10	𝑃−1	𝑃−1	ADV
m-904	43	11	gives	give	VERB
m-904	43	12	𝑃−1𝑃𝑥′	𝑃−1𝑃𝑥′	PROPN
m-904	43	13	=	=	PUNCT
m-904	44	1	𝑃−1	𝑃−1	X
m-904	44	2	[	[	PUNCT
m-904	44	3	−𝐾	−𝐾	PROPN
m-904	44	4	+	+	CCONJ
m-904	44	5	𝐶	𝐶	PROPN
m-904	44	6	0	0	PUNCT
m-904	44	7	𝐾	𝐾	PROPN
m-904	44	8	−𝐾	−𝐾	VERB
m-904	44	9	]	]	PUNCT
m-904	45	1	𝑃𝑥	𝑃𝑥	PROPN
m-904	45	2	(	(	PUNCT
m-904	45	3	15	15	NUM
m-904	45	4	)	)	PUNCT
m-904	45	5	incorporate	incorporate	NOUN
m-904	45	6	equations	equation	NOUN
m-904	45	7	(	(	PUNCT
m-904	45	8	9	9	NUM
m-904	45	9	)	)	PUNCT
m-904	45	10	,	,	PUNCT
m-904	45	11	(	(	PUNCT
m-904	45	12	11	11	NUM
m-904	45	13	)	)	PUNCT
m-904	45	14	and	and	CCONJ
m-904	45	15	(	(	PUNCT
m-904	45	16	12	12	NUM
m-904	45	17	)	)	PUNCT
m-904	45	18	into	into	ADP
m-904	45	19	(	(	PUNCT
m-904	45	20	15	15	NUM
m-904	45	21	)	)	PUNCT
m-904	45	22	,	,	PUNCT
m-904	45	23	we	we	PRON
m-904	45	24	have	have	VERB
m-904	45	25	[	[	PUNCT
m-904	45	26	𝑥1	𝑥1	NOUN
m-904	45	27	′	′	NUM
m-904	45	28	(	(	PUNCT
m-904	45	29	𝑡	𝑡	NOUN
m-904	45	30	)	)	PUNCT
m-904	45	31	𝑥2	𝑥2	NOUN
m-904	45	32	′	′	NUM
m-904	45	33	(	(	PUNCT
m-904	45	34	𝑡	𝑡	NOUN
m-904	45	35	)	)	PUNCT
m-904	45	36	]	]	PUNCT
m-904	46	1	=	=	PUNCT
m-904	46	2	[	[	PUNCT
m-904	46	3	𝜆1	𝜆1	NOUN
m-904	46	4	0	0	NUM
m-904	46	5	0	0	NUM
m-904	46	6	𝜆2	𝜆2	NOUN
m-904	46	7	]	]	PUNCT
m-904	46	8	[	[	PUNCT
m-904	46	9	𝑥1(𝑡	𝑥1(𝑡	NOUN
m-904	46	10	)	)	PUNCT
m-904	46	11	𝑥2(𝑡	𝑥2(𝑡	NOUN
m-904	46	12	)	)	PUNCT
m-904	46	13	]	]	PUNCT
m-904	46	14	(	(	PUNCT
m-904	46	15	16a	16a	NOUN
m-904	46	16	)	)	PUNCT
m-904	46	17	which	which	PRON
m-904	46	18	gives	give	VERB
m-904	46	19	𝑥1	𝑥1	NOUN
m-904	46	20	′	′	NOUN
m-904	46	21	=	=	SYM
m-904	46	22	𝜆1𝑥1	𝜆1𝑥1	X
m-904	46	23	(	(	PUNCT
m-904	46	24	16b	16b	ADJ
m-904	46	25	)	)	PUNCT
m-904	46	26	𝑥2	𝑥2	NOUN
m-904	46	27	′	′	NUM
m-904	46	28	=	=	SYM
m-904	46	29	𝜆2𝑥2	𝜆2𝑥2	X
m-904	46	30	(	(	PUNCT
m-904	46	31	16c	16c	NOUN
m-904	46	32	)	)	PUNCT
m-904	46	33	the	the	DET
m-904	46	34	system	system	NOUN
m-904	46	35	is	be	AUX
m-904	46	36	now	now	ADV
m-904	46	37	decoupled	decouple	VERB
m-904	46	38	[	[	X
m-904	46	39	9	9	NUM
m-904	46	40	,	,	PUNCT
m-904	46	41	10	10	NUM
m-904	46	42	]	]	PUNCT
m-904	46	43	.	.	PUNCT
m-904	47	1	2.4	2.4	NUM
m-904	47	2	solution	solution	NOUN
m-904	47	3	to	to	PART
m-904	47	4	example	example	VERB
m-904	47	5	the	the	DET
m-904	47	6	solutions	solution	NOUN
m-904	47	7	to	to	ADP
m-904	47	8	equation	equation	NOUN
m-904	47	9	(	(	PUNCT
m-904	47	10	16	16	NUM
m-904	47	11	)	)	PUNCT
m-904	47	12	are	be	AUX
m-904	47	13	easily	easily	ADV
m-904	47	14	obtained	obtain	VERB
m-904	47	15	using	use	VERB
m-904	47	16	separation	separation	NOUN
m-904	47	17	of	of	ADP
m-904	47	18	variables	variable	NOUN
m-904	47	19	method	method	NOUN
m-904	47	20	as	as	ADP
m-904	47	21	𝑥1(𝑡	𝑥1(𝑡	NOUN
m-904	47	22	)	)	PUNCT
m-904	47	23	=	=	SYM
m-904	47	24	𝐷1𝑒−𝐾𝑡	𝐷1𝑒−𝐾𝑡	X
m-904	47	25	(	(	PUNCT
m-904	47	26	17a	17a	NOUN
m-904	47	27	)	)	PUNCT
m-904	48	1	𝑥2(𝑡	𝑥2(𝑡	NOUN
m-904	48	2	)	)	PUNCT
m-904	48	3	=	=	SYM
m-904	48	4	𝐷2𝑒−(𝐾−𝐶)𝑡	𝐷2𝑒−(𝐾−𝐶)𝑡	X
m-904	48	5	(	(	PUNCT
m-904	48	6	17b	17b	NUM
m-904	48	7	)	)	PUNCT
m-904	48	8	ijo	ijo	PROPN
m-904	48	9	international	international	PROPN
m-904	48	10	journal	journal	PROPN
m-904	48	11	of	of	ADP
m-904	48	12	mathematics	mathematics	PROPN
m-904	48	13	(	(	PUNCT
m-904	48	14	issn	issn	PROPN
m-904	48	15	:	:	PUNCT
m-904	48	16	2992	2992	NUM
m-904	48	17	-	-	SYM
m-904	48	18	4421	4421	NUM
m-904	48	19	)	)	PUNCT
m-904	48	20	ijo	ijo	PROPN
m-904	48	21	journals	journal	NOUN
m-904	48	22	volume	volume	NOUN
m-904	48	23	07	07	NUM
m-904	48	24	|	|	NOUN
m-904	48	25	issue	issue	NOUN
m-904	48	26	07	07	NUM
m-904	48	27	|	|	CCONJ
m-904	48	28	july	july	PROPN
m-904	48	29	2024	2024	NUM
m-904	48	30	|	|	ADV
m-904	48	31	https://ijojournals.com/index.php/m/index	https://ijojournals.com/index.php/m/index	NOUN
m-904	48	32	5	5	NUM
m-904	48	33	since	since	SCONJ
m-904	48	34	𝑚	𝑚	NOUN
m-904	48	35	=	=	SYM
m-904	48	36	𝑃𝑥	𝑃𝑥	PROPN
m-904	48	37	,	,	PUNCT
m-904	48	38	we	we	PRON
m-904	48	39	have	have	VERB
m-904	48	40	[	[	PUNCT
m-904	48	41	𝑚1	𝑚1	NOUN
m-904	48	42	𝑚2	𝑚2	NOUN
m-904	48	43	]	]	PUNCT
m-904	49	1	=	=	PUNCT
m-904	49	2	[	[	PUNCT
m-904	49	3	0	0	NUM
m-904	49	4	𝐶	𝐶	PROPN
m-904	49	5	𝐾	𝐾	PROPN
m-904	49	6	1	1	NUM
m-904	49	7	1	1	NUM
m-904	49	8	]	]	PUNCT
m-904	49	9	[	[	PUNCT
m-904	49	10	𝐷1𝑒−𝐾𝑡	𝐷1𝑒−𝐾𝑡	X
m-904	49	11	𝐷2𝑒−(𝐾−𝐶)𝑡	𝐷2𝑒−(𝐾−𝐶)𝑡	X
m-904	49	12	]	]	PUNCT
m-904	49	13	.	.	PUNCT
m-904	50	1	(	(	PUNCT
m-904	50	2	18	18	NUM
m-904	50	3	)	)	PUNCT
m-904	50	4	with	with	ADP
m-904	50	5	the	the	DET
m-904	50	6	initial	initial	ADJ
m-904	50	7	conditions	condition	NOUN
m-904	50	8	𝑚1(0	𝑚1(0	PROPN
m-904	50	9	)	)	PUNCT
m-904	50	10	=	=	SYM
m-904	50	11	𝑎	𝑎	NOUN
m-904	50	12	and	and	CCONJ
m-904	50	13	𝑚2(0	𝑚2(0	ADV
m-904	50	14	)	)	PUNCT
m-904	50	15	=	=	SYM
m-904	51	1	𝑏	𝑏	NOUN
m-904	51	2	,	,	PUNCT
m-904	51	3	it	it	PRON
m-904	51	4	is	be	AUX
m-904	51	5	easy	easy	ADJ
m-904	51	6	to	to	PART
m-904	51	7	verify	verify	VERB
m-904	51	8	that	that	SCONJ
m-904	51	9	the	the	DET
m-904	51	10	solutions	solution	NOUN
m-904	51	11	to	to	ADP
m-904	51	12	the	the	DET
m-904	51	13	ivp	ivp	NOUN
m-904	51	14	in	in	ADP
m-904	51	15	(	(	PUNCT
m-904	51	16	3	3	X
m-904	51	17	)	)	PUNCT
m-904	51	18	are	be	AUX
m-904	51	19	𝑚1(𝑡	𝑚1(𝑡	PRON
m-904	51	20	)	)	PUNCT
m-904	52	1	=	=	SYM
m-904	52	2	𝑎𝑒−(𝐾−𝐶)𝑡	𝑎𝑒−(𝐾−𝐶)𝑡	PROPN
m-904	52	3	(	(	PUNCT
m-904	52	4	19a	19a	NUM
m-904	52	5	)	)	PUNCT
m-904	52	6	𝑚2(𝑡	𝑚2(𝑡	NOUN
m-904	52	7	)	)	PUNCT
m-904	52	8	=	=	SYM
m-904	52	9	(	(	PUNCT
m-904	52	10	𝑏	𝑏	DET
m-904	52	11	−	−	PROPN
m-904	52	12	𝐾𝑎	𝐾𝑎	PROPN
m-904	52	13	𝐶	𝐶	PROPN
m-904	52	14	)	)	PUNCT
m-904	52	15	𝑒−𝐾𝑡	𝑒−𝐾𝑡	PUNCT
m-904	53	1	+	+	CCONJ
m-904	53	2	𝐾𝑎	𝐾𝑎	PROPN
m-904	53	3	𝐶	𝐶	PROPN
m-904	53	4	𝑒−(𝐾−𝐶)𝑡	𝑒−(𝐾−𝐶)𝑡	PROPN
m-904	53	5	(	(	PUNCT
m-904	53	6	19b	19b	NOUN
m-904	53	7	)	)	PUNCT
m-904	53	8	3	3	NUM
m-904	53	9	existing	exist	VERB
m-904	53	10	solution	solution	NOUN
m-904	53	11	methods	method	NOUN
m-904	53	12	there	there	PRON
m-904	53	13	are	be	VERB
m-904	53	14	several	several	ADJ
m-904	53	15	methods	method	NOUN
m-904	53	16	to	to	PART
m-904	53	17	solve	solve	VERB
m-904	53	18	simultaneous	simultaneous	ADJ
m-904	53	19	differential	differential	ADJ
m-904	53	20	equations	equation	NOUN
m-904	53	21	,	,	PUNCT
m-904	53	22	each	each	PRON
m-904	53	23	has	have	VERB
m-904	53	24	it	it	PRON
m-904	53	25	own	own	VERB
m-904	53	26	restrictions	restriction	NOUN
m-904	53	27	on	on	ADP
m-904	53	28	the	the	DET
m-904	53	29	form	form	NOUN
m-904	53	30	of	of	ADP
m-904	53	31	the	the	DET
m-904	53	32	equations	equation	NOUN
m-904	53	33	.	.	PUNCT
m-904	54	1	[	[	X
m-904	54	2	11	11	NUM
m-904	54	3	,	,	PUNCT
m-904	54	4	12	12	NUM
m-904	54	5	,	,	PUNCT
m-904	54	6	13	13	NUM
m-904	54	7	,	,	PUNCT
m-904	54	8	14	14	NUM
m-904	54	9	,	,	PUNCT
m-904	54	10	15	15	NUM
m-904	54	11	,	,	PUNCT
m-904	54	12	16	16	NUM
m-904	54	13	]	]	PUNCT
m-904	54	14	these	these	DET
m-904	54	15	methods	method	NOUN
m-904	54	16	are	be	AUX
m-904	54	17	:	:	PUNCT
m-904	54	18	(	(	PUNCT
m-904	54	19	i	i	NOUN
m-904	54	20	)	)	PUNCT
m-904	54	21	substitution	substitution	NOUN
m-904	54	22	(	(	PUNCT
m-904	54	23	ii	ii	NOUN
m-904	54	24	)	)	PUNCT
m-904	54	25	elimination	elimination	NOUN
m-904	54	26	(	(	PUNCT
m-904	54	27	iii	iii	NOUN
m-904	54	28	)	)	PUNCT
m-904	54	29	d	d	NOUN
m-904	54	30	-	-	NOUN
m-904	54	31	operator	operator	NOUN
m-904	54	32	(	(	PUNCT
m-904	54	33	iv	iv	X
m-904	54	34	)	)	PUNCT
m-904	54	35	laplace	laplace	NOUN
m-904	54	36	transform	transform	NOUN
m-904	54	37	(	(	PUNCT
m-904	54	38	v	v	NOUN
m-904	54	39	)	)	PUNCT
m-904	54	40	numerical	numerical	ADJ
m-904	54	41	recursive	recursive	ADJ
m-904	54	42	calculations	calculation	NOUN
m-904	54	43	(	(	PUNCT
m-904	54	44	vi	vi	NOUN
m-904	54	45	)	)	PUNCT
m-904	54	46	software	software	NOUN
m-904	54	47	package	package	NOUN
m-904	54	48	ijo	ijo	PROPN
m-904	54	49	international	international	PROPN
m-904	54	50	journal	journal	PROPN
m-904	54	51	of	of	ADP
m-904	54	52	mathematics	mathematics	PROPN
m-904	54	53	(	(	PUNCT
m-904	54	54	issn	issn	PROPN
m-904	54	55	:	:	PUNCT
m-904	54	56	2992	2992	NUM
m-904	54	57	-	-	SYM
m-904	54	58	4421	4421	NUM
m-904	54	59	)	)	PUNCT
m-904	54	60	ijo	ijo	PROPN
m-904	54	61	journals	journal	NOUN
m-904	54	62	volume	volume	NOUN
m-904	54	63	07	07	NUM
m-904	54	64	|	|	NOUN
m-904	54	65	issue	issue	NOUN
m-904	54	66	07	07	NUM
m-904	55	1	|	|	CCONJ
m-904	55	2	july	july	PROPN
m-904	55	3	2024	2024	NUM
m-904	55	4	|	|	ADV
m-904	55	5	https://ijojournals.com/index.php/m/index	https://ijojournals.com/index.php/m/index	NOUN
m-904	55	6	6	6	NUM
m-904	55	7	4	4	NUM
m-904	55	8	.	.	PUNCT
m-904	55	9	proposed	propose	VERB
m-904	55	10	algebraic	algebraic	ADJ
m-904	55	11	solutions	solution	NOUN
m-904	55	12	a	a	DET
m-904	55	13	special	special	ADJ
m-904	55	14	case	case	NOUN
m-904	55	15	of	of	ADP
m-904	55	16	simultaneous	simultaneous	ADJ
m-904	55	17	differential	differential	ADJ
m-904	55	18	equations	equation	NOUN
m-904	55	19	is	be	AUX
m-904	55	20	focused	focus	VERB
m-904	55	21	here	here	ADV
m-904	55	22	,	,	PUNCT
m-904	55	23	where	where	SCONJ
m-904	55	24	only	only	ADV
m-904	55	25	one	one	NUM
m-904	55	26	dependent	dependent	ADJ
m-904	55	27	variable	variable	ADJ
m-904	55	28	y	y	NOUN
m-904	55	29	,	,	PUNCT
m-904	55	30	and	and	CCONJ
m-904	55	31	one	one	NUM
m-904	55	32	independent	independent	ADJ
m-904	55	33	variable	variable	NOUN
m-904	55	34	x	x	NOUN
m-904	55	35	,	,	PUNCT
m-904	55	36	are	be	AUX
m-904	55	37	involved	involve	VERB
m-904	55	38	.	.	PUNCT
m-904	56	1	however	however	ADV
m-904	56	2	,	,	PUNCT
m-904	56	3	the	the	DET
m-904	56	4	solution	solution	NOUN
m-904	56	5	must	must	AUX
m-904	56	6	satisfy	satisfy	VERB
m-904	56	7	two	two	NUM
m-904	56	8	non	non	ADJ
m-904	56	9	-	-	ADJ
m-904	56	10	homogeneous	homogeneous	ADJ
m-904	56	11	differential	differential	ADJ
m-904	56	12	equations	equation	NOUN
m-904	56	13	simultaneously	simultaneously	ADV
m-904	56	14	.	.	PUNCT
m-904	57	1	the	the	DET
m-904	57	2	main	main	ADJ
m-904	57	3	result	result	NOUN
m-904	57	4	of	of	ADP
m-904	57	5	this	this	DET
m-904	57	6	paper	paper	NOUN
m-904	57	7	is	be	AUX
m-904	57	8	stated	state	VERB
m-904	57	9	as	as	ADP
m-904	57	10	a	a	DET
m-904	57	11	theorem	theorem	NOUN
m-904	57	12	below	below	ADV
m-904	57	13	.	.	PUNCT
m-904	58	1	theorem	theorem	VERB
m-904	58	2	1	1	NUM
m-904	58	3	the	the	DET
m-904	58	4	solutions	solution	NOUN
m-904	58	5	to	to	ADP
m-904	58	6	the	the	DET
m-904	58	7	set	set	NOUN
m-904	58	8	of	of	ADP
m-904	58	9	non	non	ADJ
m-904	58	10	-	-	ADJ
m-904	58	11	homogeneous	homogeneous	ADJ
m-904	58	12	linear	linear	ADJ
m-904	58	13	first	first	ADJ
m-904	58	14	-	-	PUNCT
m-904	58	15	order	order	NOUN
m-904	58	16	simultaneous	simultaneous	ADJ
m-904	58	17	ordinary	ordinary	ADJ
m-904	58	18	differential	differential	ADJ
m-904	58	19	equations	equation	NOUN
m-904	58	20	with	with	ADP
m-904	58	21	constant	constant	ADJ
m-904	58	22	coefficients	coefficient	NOUN
m-904	58	23	in	in	ADP
m-904	58	24	the	the	DET
m-904	58	25	form	form	NOUN
m-904	58	26	of	of	ADP
m-904	58	27	𝑎1𝑦′	𝑎1𝑦′	NOUN
m-904	58	28	+	+	CCONJ
m-904	58	29	𝑎0𝑦	𝑎0𝑦	NOUN
m-904	58	30	=	=	PUNCT
m-904	58	31	𝑓(𝑥	𝑓(𝑥	NOUN
m-904	58	32	)	)	PUNCT
m-904	58	33	(	(	PUNCT
m-904	58	34	20a	20a	NOUN
m-904	58	35	)	)	PUNCT
m-904	58	36	𝑏1𝑦′	𝑏1𝑦′	NOUN
m-904	59	1	+	+	CCONJ
m-904	59	2	𝑏0𝑦	𝑏0𝑦	NOUN
m-904	59	3	=	=	SYM
m-904	59	4	𝑔(𝑥	𝑔(𝑥	PROPN
m-904	59	5	)	)	PUNCT
m-904	59	6	(	(	PUNCT
m-904	59	7	20b	20b	NOUN
m-904	59	8	)	)	PUNCT
m-904	59	9	are	be	AUX
m-904	59	10	given	give	VERB
m-904	59	11	by	by	ADP
m-904	59	12	𝑦(𝑥	𝑦(𝑥	NOUN
m-904	59	13	)	)	PUNCT
m-904	59	14	=	=	PUNCT
m-904	59	15	𝑔(𝑥)𝑎1−𝑓(𝑥)𝑏1	𝑔(𝑥)𝑎1−𝑓(𝑥)𝑏1	NUM
m-904	59	16	𝑎1𝑏0−𝑎0𝑏1	𝑎1𝑏0−𝑎0𝑏1	PROPN
m-904	59	17	(	(	PUNCT
m-904	59	18	21a	21a	NUM
m-904	59	19	)	)	PUNCT
m-904	59	20	𝑦′(𝑥	𝑦′(𝑥	NOUN
m-904	59	21	)	)	PUNCT
m-904	59	22	=	=	SYM
m-904	59	23	𝑓(𝑥)𝑏0−𝑔(𝑥)𝑎0	𝑓(𝑥)𝑏0−𝑔(𝑥)𝑎0	SYM
m-904	59	24	𝑎1𝑏0−𝑎0𝑏1	𝑎1𝑏0−𝑎0𝑏1	PROPN
m-904	59	25	(	(	PUNCT
m-904	59	26	21b	21b	NOUN
m-904	59	27	)	)	PUNCT
m-904	59	28	with	with	ADP
m-904	59	29	the	the	DET
m-904	59	30	necessary	necessary	ADJ
m-904	59	31	condition	condition	NOUN
m-904	59	32	𝑎1𝑔′(𝑥	𝑎1𝑔′(𝑥	PROPN
m-904	59	33	)	)	PUNCT
m-904	59	34	−	−	PROPN
m-904	59	35	𝑏1𝑓′(𝑥	𝑏1𝑓′(𝑥	NOUN
m-904	59	36	)	)	PUNCT
m-904	60	1	=	=	SYM
m-904	60	2	𝑏0𝑓(𝑥	𝑏0𝑓(𝑥	NOUN
m-904	60	3	)	)	PUNCT
m-904	60	4	−	−	PROPN
m-904	60	5	𝑎0𝑔(𝑥	𝑎0𝑔(𝑥	PROPN
m-904	60	6	)	)	PUNCT
m-904	60	7	.	.	PUNCT
m-904	61	1	(	(	PUNCT
m-904	61	2	21c	21c	NUM
m-904	61	3	)	)	PUNCT
m-904	61	4	proof	proof	NOUN
m-904	61	5	write	write	VERB
m-904	61	6	the	the	DET
m-904	61	7	given	give	VERB
m-904	61	8	equations	equation	NOUN
m-904	61	9	in	in	ADP
m-904	61	10	matric	matric	NOUN
m-904	61	11	form	form	NOUN
m-904	61	12	as	as	SCONJ
m-904	61	13	[	[	PUNCT
m-904	61	14	𝑎1	𝑎1	NOUN
m-904	61	15	𝑎0	𝑎0	AUX
m-904	61	16	𝑏1	𝑏1	VERB
m-904	61	17	𝑏0	𝑏0	NOUN
m-904	61	18	]	]	PUNCT
m-904	61	19	[	[	PUNCT
m-904	61	20	𝑦′	𝑦′	X
m-904	61	21	𝑦	𝑦	NOUN
m-904	61	22	]	]	X
m-904	61	23	=	=	PUNCT
m-904	61	24	[	[	PUNCT
m-904	61	25	𝑓(𝑥	𝑓(𝑥	NOUN
m-904	61	26	)	)	PUNCT
m-904	61	27	𝑔(𝑥	𝑔(𝑥	PROPN
m-904	61	28	)	)	PUNCT
m-904	61	29	]	]	PUNCT
m-904	62	1	(	(	PUNCT
m-904	62	2	22	22	NUM
m-904	62	3	)	)	PUNCT
m-904	62	4	define	define	VERB
m-904	62	5	the	the	DET
m-904	62	6	following	follow	VERB
m-904	62	7	three	three	NUM
m-904	62	8	determinants	determinant	NOUN
m-904	62	9	|𝐴|	|𝐴|	PART
m-904	63	1	=	=	SYM
m-904	63	2	|	|	ADV
m-904	63	3	𝑎1	𝑎1	INTJ
m-904	63	4	𝑎0	𝑎0	AUX
m-904	63	5	𝑏1	𝑏1	VERB
m-904	63	6	𝑏0	𝑏0	NOUN
m-904	63	7	|	|	ADV
m-904	63	8	=	=	PUNCT
m-904	63	9	𝑎1𝑏0	𝑎1𝑏0	VERB
m-904	63	10	−	−	PROPN
m-904	63	11	𝑎0𝑏1	𝑎0𝑏1	PUNCT
m-904	63	12	(	(	PUNCT
m-904	63	13	23a	23a	NUM
m-904	63	14	)	)	PUNCT
m-904	63	15	|𝐴1|	|𝐴1|	NOUN
m-904	63	16	=	=	PUNCT
m-904	63	17	|	|	ADV
m-904	63	18	𝑓(𝑥	𝑓(𝑥	NOUN
m-904	63	19	)	)	PUNCT
m-904	63	20	𝑎0	𝑎0	ADV
m-904	63	21	𝑔(𝑥	𝑔(𝑥	NOUN
m-904	63	22	)	)	PUNCT
m-904	63	23	𝑏0	𝑏0	NOUN
m-904	63	24	|	|	NOUN
m-904	63	25	=	=	SYM
m-904	63	26	𝑏0𝑓(𝑥	𝑏0𝑓(𝑥	PROPN
m-904	63	27	)	)	PUNCT
m-904	63	28	−	−	PROPN
m-904	63	29	𝑎0𝑔(𝑥	𝑎0𝑔(𝑥	PROPN
m-904	63	30	)	)	PUNCT
m-904	63	31	(	(	PUNCT
m-904	63	32	23b	23b	NOUN
m-904	63	33	)	)	PUNCT
m-904	63	34	|𝐴2|	|𝐴2|	PROPN
m-904	63	35	=	=	PUNCT
m-904	64	1	|	|	ADV
m-904	64	2	𝑎1	𝑎1	ADJ
m-904	64	3	𝑓(𝑥	𝑓(𝑥	NOUN
m-904	64	4	)	)	PUNCT
m-904	64	5	𝑏1	𝑏1	NOUN
m-904	64	6	𝑔(𝑥	𝑔(𝑥	NUM
m-904	64	7	)	)	PUNCT
m-904	64	8	|	|	ADV
m-904	64	9	=	=	SYM
m-904	64	10	𝑎1𝑔(𝑥	𝑎1𝑔(𝑥	NOUN
m-904	64	11	)	)	PUNCT
m-904	64	12	−	−	NOUN
m-904	64	13	𝑏1𝑓(𝑥	𝑏1𝑓(𝑥	NOUN
m-904	64	14	)	)	PUNCT
m-904	64	15	(	(	PUNCT
m-904	64	16	23c	23c	NOUN
m-904	64	17	)	)	PUNCT
m-904	64	18	using	use	VERB
m-904	64	19	cramer	cramer	PROPN
m-904	64	20	’s	’s	PART
m-904	64	21	rule	rule	NOUN
m-904	64	22	,	,	PUNCT
m-904	64	23	the	the	DET
m-904	64	24	solutions	solution	NOUN
m-904	64	25	are	be	AUX
m-904	64	26	𝑦′(𝑥	𝑦′(𝑥	ADJ
m-904	64	27	)	)	PUNCT
m-904	64	28	=	=	SYM
m-904	64	29	𝑏0𝑓(𝑥)−𝑎0𝑔(𝑥	𝑏0𝑓(𝑥)−𝑎0𝑔(𝑥	ADJ
m-904	64	30	)	)	PUNCT
m-904	64	31	𝑎1𝑏0−𝑎0𝑏1	𝑎1𝑏0−𝑎0𝑏1	PROPN
m-904	64	32	and	and	CCONJ
m-904	64	33	𝑦(𝑥	𝑦(𝑥	PROPN
m-904	64	34	)	)	PUNCT
m-904	64	35	=	=	SYM
m-904	64	36	𝑎1𝑔(𝑥)−𝑏1𝑓(𝑥	𝑎1𝑔(𝑥)−𝑏1𝑓(𝑥	NOUN
m-904	64	37	)	)	PUNCT
m-904	64	38	𝑎1𝑏0−𝑎0𝑏1	𝑎1𝑏0−𝑎0𝑏1	PROPN
m-904	64	39	(	(	PUNCT
m-904	64	40	24	24	NUM
m-904	64	41	)	)	PUNCT
m-904	64	42	however	however	ADV
m-904	64	43	,	,	PUNCT
m-904	64	44	since	since	SCONJ
m-904	64	45	𝑦(𝑥	𝑦(𝑥	NOUN
m-904	64	46	)	)	PUNCT
m-904	64	47	and	and	CCONJ
m-904	64	48	𝑦′(𝑥	𝑦′(𝑥	VERB
m-904	64	49	)	)	PUNCT
m-904	64	50	are	be	AUX
m-904	64	51	not	not	PART
m-904	64	52	constants	constant	NOUN
m-904	64	53	and	and	CCONJ
m-904	64	54	𝑦′(𝑥	𝑦′(𝑥	NOUN
m-904	64	55	)	)	PUNCT
m-904	64	56	is	be	AUX
m-904	64	57	obtained	obtain	VERB
m-904	64	58	by	by	ADP
m-904	64	59	differentiating	differentiate	VERB
m-904	64	60	𝑦(𝑥	𝑦(𝑥	NOUN
m-904	64	61	)	)	PUNCT
m-904	64	62	with	with	ADP
m-904	64	63	respect	respect	NOUN
m-904	64	64	to	to	ADP
m-904	64	65	x	x	PRON
m-904	64	66	,	,	PUNCT
m-904	64	67	the	the	DET
m-904	64	68	following	follow	VERB
m-904	64	69	necessary	necessary	ADJ
m-904	64	70	condition	condition	NOUN
m-904	64	71	must	must	AUX
m-904	64	72	be	be	AUX
m-904	64	73	satisfied	satisfied	ADJ
m-904	64	74	:	:	PUNCT
m-904	64	75	𝑎1𝑔′(𝑥	𝑎1𝑔′(𝑥	ADJ
m-904	64	76	)	)	PUNCT
m-904	65	1	−	−	PROPN
m-904	65	2	𝑏1𝑓′(𝑥	𝑏1𝑓′(𝑥	NOUN
m-904	65	3	)	)	PUNCT
m-904	65	4	=	=	SYM
m-904	65	5	𝑏0𝑓(𝑥	𝑏0𝑓(𝑥	NOUN
m-904	65	6	)	)	PUNCT
m-904	65	7	−	−	PROPN
m-904	65	8	𝑎0𝑔(𝑥	𝑎0𝑔(𝑥	PROPN
m-904	65	9	)	)	PUNCT
m-904	66	1	∎	∎	PROPN
m-904	66	2	(	(	PUNCT
m-904	66	3	25	25	NUM
m-904	66	4	)	)	PUNCT
m-904	66	5	note	note	NOUN
m-904	66	6	that	that	SCONJ
m-904	66	7	the	the	DET
m-904	66	8	solution	solution	NOUN
m-904	66	9	𝑦(𝑥	𝑦(𝑥	NOUN
m-904	66	10	)	)	PUNCT
m-904	66	11	can	can	AUX
m-904	66	12	be	be	AUX
m-904	66	13	obtained	obtain	VERB
m-904	66	14	using	use	VERB
m-904	66	15	simple	simple	ADJ
m-904	66	16	algebraic	algebraic	ADJ
m-904	66	17	computation	computation	NOUN
m-904	66	18	.	.	PUNCT
m-904	67	1	now	now	ADV
m-904	67	2	let	let	VERB
m-904	67	3	’s	’s	NOUN
m-904	67	4	look	look	VERB
m-904	67	5	at	at	ADP
m-904	67	6	the	the	DET
m-904	67	7	four	four	NUM
m-904	67	8	common	common	ADJ
m-904	67	9	functions	function	NOUN
m-904	67	10	for	for	ADP
m-904	67	11	𝑓(𝑥	𝑓(𝑥	NOUN
m-904	67	12	)	)	PUNCT
m-904	67	13	and	and	CCONJ
m-904	67	14	𝑔(𝑥	𝑔(𝑥	NUM
m-904	67	15	)	)	PUNCT
m-904	67	16	.	.	PUNCT
m-904	68	1	4.1	4.1	NUM
m-904	68	2	case	case	NOUN
m-904	68	3	1	1	NUM
m-904	68	4	:	:	PUNCT
m-904	68	5	𝒇(𝒙	𝒇(𝒙	NUM
m-904	68	6	)	)	PUNCT
m-904	68	7	𝐚𝐧𝐝	𝐚𝐧𝐝	CCONJ
m-904	68	8	𝒈(𝒙	𝒈(𝒙	PROPN
m-904	68	9	)	)	PUNCT
m-904	68	10	are	be	AUX
m-904	68	11	constants	constant	NOUN
m-904	68	12	.	.	PUNCT
m-904	69	1	ijo	ijo	PROPN
m-904	69	2	international	international	PROPN
m-904	69	3	journal	journal	PROPN
m-904	69	4	of	of	ADP
m-904	69	5	mathematics	mathematics	PROPN
m-904	69	6	(	(	PUNCT
m-904	69	7	issn	issn	PROPN
m-904	69	8	:	:	PUNCT
m-904	69	9	2992	2992	NUM
m-904	69	10	-	-	SYM
m-904	69	11	4421	4421	NUM
m-904	69	12	)	)	PUNCT
m-904	69	13	ijo	ijo	PROPN
m-904	69	14	journals	journal	NOUN
m-904	69	15	volume	volume	NOUN
m-904	69	16	07	07	NUM
m-904	69	17	|	|	NOUN
m-904	69	18	issue	issue	NOUN
m-904	69	19	07	07	NUM
m-904	70	1	|	|	CCONJ
m-904	70	2	july	july	PROPN
m-904	70	3	2024	2024	NUM
m-904	70	4	|	|	ADV
m-904	70	5	https://ijojournals.com/index.php/m/index	https://ijojournals.com/index.php/m/index	NOUN
m-904	70	6	7	7	NUM
m-904	70	7	𝑓(𝑥	𝑓(𝑥	NOUN
m-904	70	8	)	)	PUNCT
m-904	70	9	=	=	SYM
m-904	70	10	𝑓	𝑓	PROPN
m-904	70	11	and	and	CCONJ
m-904	70	12	𝑔(𝑥	𝑔(𝑥	NUM
m-904	70	13	)	)	PUNCT
m-904	70	14	=	=	SYM
m-904	71	1	𝑔	𝑔	NOUN
m-904	71	2	,	,	PUNCT
m-904	71	3	where	where	SCONJ
m-904	71	4	f	f	PROPN
m-904	71	5	and	and	CCONJ
m-904	71	6	g	g	PROPN
m-904	71	7	are	be	AUX
m-904	71	8	scalar	scalar	ADJ
m-904	71	9	constants	constant	NOUN
m-904	71	10	.	.	PUNCT
m-904	72	1	the	the	DET
m-904	72	2	necessary	necessary	ADJ
m-904	72	3	condition	condition	NOUN
m-904	72	4	becomes	become	VERB
m-904	72	5	0	0	NUM
m-904	73	1	=	=	SYM
m-904	73	2	𝑏0𝑓	𝑏0𝑓	PROPN
m-904	73	3	−	−	PROPN
m-904	73	4	𝑎0𝑔	𝑎0𝑔	NOUN
m-904	73	5	(	(	PUNCT
m-904	73	6	26	26	NUM
m-904	73	7	)	)	PUNCT
m-904	73	8	and	and	CCONJ
m-904	73	9	the	the	DET
m-904	73	10	solutions	solution	NOUN
m-904	73	11	are	be	AUX
m-904	73	12	𝑦′(𝑥	𝑦′(𝑥	ADJ
m-904	73	13	)	)	PUNCT
m-904	73	14	=	=	PRON
m-904	73	15	𝑏0𝑓−𝑎0𝑔	𝑏0𝑓−𝑎0𝑔	VERB
m-904	73	16	𝛼	𝛼	NOUN
m-904	73	17	=	=	NOUN
m-904	73	18	0	0	NUM
m-904	73	19	(	(	PUNCT
m-904	73	20	27a	27a	NUM
m-904	73	21	)	)	PUNCT
m-904	73	22	𝑦(𝑥	𝑦(𝑥	NOUN
m-904	73	23	)	)	PUNCT
m-904	73	24	=	=	PUNCT
m-904	74	1	𝑎1𝑔−𝑏1𝑓	𝑎1𝑔−𝑏1𝑓	PROPN
m-904	74	2	𝛼	𝛼	PROPN
m-904	74	3	(	(	PUNCT
m-904	74	4	27b	27b	NUM
m-904	74	5	)	)	PUNCT
m-904	74	6	where	where	SCONJ
m-904	74	7	the	the	DET
m-904	74	8	constant	constant	ADJ
m-904	74	9	𝛼	𝛼	NOUN
m-904	74	10	is	be	AUX
m-904	74	11	defined	define	VERB
m-904	74	12	as	as	ADP
m-904	74	13	𝛼	𝛼	NOUN
m-904	74	14	=	=	PUNCT
m-904	74	15	𝑎1𝑏0	𝑎1𝑏0	PUNCT
m-904	74	16	−	−	PROPN
m-904	74	17	𝑎0𝑏1	𝑎0𝑏1	PUNCT
m-904	74	18	(	(	PUNCT
m-904	74	19	28	28	NUM
m-904	74	20	)	)	PUNCT
m-904	74	21	numerical	numerical	ADJ
m-904	74	22	example	example	NOUN
m-904	74	23	1	1	NUM
m-904	74	24	2𝑦′	2𝑦′	NUM
m-904	75	1	+	+	NUM
m-904	75	2	3𝑦	3𝑦	NOUN
m-904	75	3	=	=	SYM
m-904	75	4	6	6	NUM
m-904	75	5	5𝑦′	5𝑦′	NUM
m-904	75	6	−	−	PROPN
m-904	75	7	𝑦	𝑦	SYM
m-904	75	8	=	=	NOUN
m-904	75	9	−2	−2	NOUN
m-904	76	1	𝑎1	𝑎1	NOUN
m-904	76	2	=	=	SYM
m-904	76	3	2	2	NUM
m-904	76	4	,	,	PUNCT
m-904	76	5	𝑎0	𝑎0	PROPN
m-904	76	6	=	=	SYM
m-904	76	7	3	3	NUM
m-904	76	8	,	,	PUNCT
m-904	76	9	𝑏1	𝑏1	ADJ
m-904	76	10	=	=	SYM
m-904	76	11	5	5	NUM
m-904	76	12	,	,	PUNCT
m-904	76	13	𝑏0	𝑏0	NOUN
m-904	76	14	=	=	SYM
m-904	76	15	−1	−1	NOUN
m-904	76	16	,	,	PUNCT
m-904	76	17	𝑓	𝑓	NOUN
m-904	76	18	=	=	SYM
m-904	76	19	6	6	NUM
m-904	76	20	and	and	CCONJ
m-904	76	21	𝑔	𝑔	PROPN
m-904	76	22	=	=	PUNCT
m-904	76	23	−2	−2	PROPN
m-904	76	24	.	.	PUNCT
m-904	77	1	𝑏0𝑓	𝑏0𝑓	PROPN
m-904	77	2	−	−	PROPN
m-904	77	3	𝑎0𝑔	𝑎0𝑔	NOUN
m-904	77	4	=	=	SYM
m-904	77	5	(	(	PUNCT
m-904	77	6	−1)(6	−1)(6	NOUN
m-904	77	7	)	)	PUNCT
m-904	77	8	−	−	PROPN
m-904	77	9	(	(	PUNCT
m-904	77	10	3)(−2	3)(−2	NUM
m-904	77	11	)	)	PUNCT
m-904	77	12	=	=	SYM
m-904	78	1	−6	−6	NOUN
m-904	79	1	+	+	CCONJ
m-904	79	2	6	6	NUM
m-904	79	3	=	=	SYM
m-904	79	4	0	0	NUM
m-904	79	5	𝛼	𝛼	NOUN
m-904	79	6	=	=	SYM
m-904	79	7	(	(	PUNCT
m-904	79	8	2)(−1	2)(−1	NUM
m-904	79	9	)	)	PUNCT
m-904	79	10	−	−	PROPN
m-904	79	11	(	(	PUNCT
m-904	79	12	3)(5	3)(5	NUM
m-904	79	13	)	)	PUNCT
m-904	79	14	=	=	SYM
m-904	80	1	−17	−17	NOUN
m-904	80	2	so	so	ADV
m-904	80	3	,	,	PUNCT
m-904	80	4	𝑦′(𝑥	𝑦′(𝑥	PROPN
m-904	80	5	)	)	PUNCT
m-904	80	6	=	=	SYM
m-904	80	7	(	(	PUNCT
m-904	80	8	−1)(6)−(3)(−2	−1)(6)−(3)(−2	NUM
m-904	80	9	)	)	PUNCT
m-904	80	10	−17	−17	X
m-904	81	1	=	=	SYM
m-904	81	2	0	0	NUM
m-904	81	3	𝑦(𝑥	𝑦(𝑥	PROPN
m-904	81	4	)	)	PUNCT
m-904	81	5	=	=	SYM
m-904	81	6	(	(	PUNCT
m-904	81	7	2)(−2)−(5)(6	2)(−2)−(5)(6	NUM
m-904	81	8	)	)	PUNCT
m-904	81	9	−17	−17	NOUN
m-904	81	10	=	=	PUNCT
m-904	81	11	−34	−34	NOUN
m-904	81	12	−17	−17	X
m-904	81	13	=	=	SYM
m-904	81	14	2	2	NUM
m-904	81	15	(	(	PUNCT
m-904	81	16	29	29	NUM
m-904	81	17	)	)	PUNCT
m-904	81	18	ijo	ijo	PROPN
m-904	81	19	international	international	PROPN
m-904	81	20	journal	journal	PROPN
m-904	81	21	of	of	ADP
m-904	81	22	mathematics	mathematics	PROPN
m-904	81	23	(	(	PUNCT
m-904	81	24	issn	issn	PROPN
m-904	81	25	:	:	PUNCT
m-904	81	26	2992	2992	NUM
m-904	81	27	-	-	SYM
m-904	81	28	4421	4421	NUM
m-904	81	29	)	)	PUNCT
m-904	81	30	ijo	ijo	PROPN
m-904	81	31	journals	journal	NOUN
m-904	81	32	volume	volume	NOUN
m-904	81	33	07	07	NUM
m-904	81	34	|	|	NOUN
m-904	81	35	issue	issue	NOUN
m-904	81	36	07	07	NUM
m-904	82	1	|	|	CCONJ
m-904	82	2	july	july	PROPN
m-904	82	3	2024	2024	NUM
m-904	82	4	|	|	ADV
m-904	82	5	https://ijojournals.com/index.php/m/index	https://ijojournals.com/index.php/m/index	NOUN
m-904	82	6	8	8	NUM
m-904	82	7	check	check	NOUN
m-904	82	8	:	:	PUNCT
m-904	82	9	eq	eq	NOUN
m-904	82	10	1	1	NUM
m-904	82	11	:	:	SYM
m-904	82	12	2𝑦′	2𝑦′	NUM
m-904	83	1	+	+	NUM
m-904	83	2	3𝑦	3𝑦	NOUN
m-904	83	3	=	=	SYM
m-904	83	4	2(0	2(0	NUM
m-904	83	5	)	)	PUNCT
m-904	84	1	+	+	NOUN
m-904	84	2	3(2	3(2	NUM
m-904	84	3	)	)	PUNCT
m-904	85	1	=	=	SYM
m-904	85	2	6	6	NUM
m-904	85	3	√	√	NUM
m-904	85	4	eq	eq	ADP
m-904	85	5	2	2	NUM
m-904	85	6	:	:	PUNCT
m-904	85	7	5𝑦′	5𝑦′	NUM
m-904	85	8	−	−	PROPN
m-904	85	9	𝑦	𝑦	SYM
m-904	85	10	=	=	SYM
m-904	85	11	5(0	5(0	NUM
m-904	85	12	)	)	PUNCT
m-904	85	13	−	−	PROPN
m-904	85	14	(	(	PUNCT
m-904	85	15	2	2	NUM
m-904	85	16	)	)	PUNCT
m-904	85	17	=	=	SYM
m-904	86	1	−2	−2	NOUN
m-904	86	2	√	√	NOUN
m-904	86	3	4.2	4.2	NUM
m-904	86	4	case	case	NOUN
m-904	86	5	2	2	NUM
m-904	86	6	:	:	PUNCT
m-904	86	7	𝒇(𝒙	𝒇(𝒙	NUM
m-904	86	8	)	)	PUNCT
m-904	86	9	𝐚𝐧𝐝	𝐚𝐧𝐝	CCONJ
m-904	86	10	𝒈(𝒙	𝒈(𝒙	PROPN
m-904	86	11	)	)	PUNCT
m-904	86	12	are	be	AUX
m-904	86	13	linear	linear	NOUN
m-904	86	14	functions	function	NOUN
m-904	86	15	of	of	ADP
m-904	86	16	x.	x.	NOUN
m-904	86	17	𝑓(𝑥	𝑓(𝑥	NOUN
m-904	86	18	)	)	PUNCT
m-904	87	1	=	=	SYM
m-904	87	2	𝑚1𝑥	𝑚1𝑥	PUNCT
m-904	88	1	+	+	NUM
m-904	88	2	𝑚0	𝑚0	NOUN
m-904	88	3	and	and	CCONJ
m-904	88	4	𝑔(𝑥	𝑔(𝑥	NUM
m-904	88	5	)	)	PUNCT
m-904	89	1	=	=	SYM
m-904	89	2	𝑛1𝑥	𝑛1𝑥	NOUN
m-904	89	3	+	+	CCONJ
m-904	89	4	𝑛0	𝑛0	VERB
m-904	89	5	.	.	PUNCT
m-904	90	1	the	the	DET
m-904	90	2	necessary	necessary	ADJ
m-904	90	3	condition	condition	NOUN
m-904	90	4	becomes	become	VERB
m-904	90	5	𝑎1𝑛1	𝑎1𝑛1	NOUN
m-904	90	6	−	−	PROPN
m-904	90	7	𝑏1𝑚1	𝑏1𝑚1	PROPN
m-904	90	8	=	=	X
m-904	90	9	𝑏0(𝑚1𝑥	𝑏0(𝑚1𝑥	PROPN
m-904	90	10	+	+	NUM
m-904	90	11	𝑚0	𝑚0	NOUN
m-904	90	12	)	)	PUNCT
m-904	90	13	−	−	PROPN
m-904	91	1	𝑎0(𝑛1𝑥	𝑎0(𝑛1𝑥	PROPN
m-904	91	2	+	+	CCONJ
m-904	91	3	𝑛0	𝑛0	VERB
m-904	91	4	)	)	PUNCT
m-904	91	5	=	=	SYM
m-904	91	6	𝑏0𝑚1𝑥	𝑏0𝑚1𝑥	PUNCT
m-904	92	1	+	+	CCONJ
m-904	92	2	𝑏0𝑚0	𝑏0𝑚0	AUX
m-904	92	3	−	−	NOUN
m-904	92	4	𝑎0𝑛1𝑥	𝑎0𝑛1𝑥	SYM
m-904	92	5	−	−	ADP
m-904	92	6	𝑎0𝑛0	𝑎0𝑛0	NOUN
m-904	92	7	=	=	PRON
m-904	92	8	(	(	PUNCT
m-904	92	9	𝑏0𝑚1	𝑏0𝑚1	X
m-904	92	10	−	−	X
m-904	92	11	𝑎0𝑛1)𝑥	𝑎0𝑛1)𝑥	NOUN
m-904	92	12	+	+	CCONJ
m-904	92	13	(	(	PUNCT
m-904	92	14	𝑏0𝑚0	𝑏0𝑚0	PROPN
m-904	92	15	−	−	NOUN
m-904	92	16	𝑎0𝑛0	𝑎0𝑛0	NOUN
m-904	92	17	)	)	PUNCT
m-904	92	18	(	(	PUNCT
m-904	92	19	30	30	X
m-904	92	20	)	)	PUNCT
m-904	92	21	equating	equate	VERB
m-904	92	22	coefficients	coefficient	NOUN
m-904	92	23	of	of	ADP
m-904	92	24	like	like	ADP
m-904	92	25	powers	power	NOUN
m-904	92	26	yield	yield	VERB
m-904	92	27	the	the	DET
m-904	92	28	following	follow	VERB
m-904	92	29	two	two	NUM
m-904	92	30	necessary	necessary	ADJ
m-904	92	31	conditions	condition	NOUN
m-904	92	32	.	.	PUNCT
m-904	93	1	𝑏0𝑚1	𝑏0𝑚1	X
m-904	94	1	−	−	NUM
m-904	94	2	𝑎0𝑛1	𝑎0𝑛1	X
m-904	94	3	=	=	SYM
m-904	94	4	0	0	NUM
m-904	94	5	(	(	PUNCT
m-904	94	6	31a	31a	NUM
m-904	94	7	)	)	PUNCT
m-904	94	8	𝑎1𝑛1	𝑎1𝑛1	VERB
m-904	95	1	−	−	PROPN
m-904	95	2	𝑏1𝑚1	𝑏1𝑚1	X
m-904	95	3	=	=	PUNCT
m-904	95	4	𝑏0𝑚0	𝑏0𝑚0	PROPN
m-904	95	5	−	−	NOUN
m-904	95	6	𝑎0𝑛0	𝑎0𝑛0	PROPN
m-904	95	7	(	(	PUNCT
m-904	95	8	31b	31b	NUM
m-904	95	9	)	)	PUNCT
m-904	95	10	the	the	DET
m-904	95	11	solutions	solution	NOUN
m-904	95	12	are	be	AUX
m-904	95	13	𝑦(𝑥	𝑦(𝑥	NOUN
m-904	95	14	)	)	PUNCT
m-904	95	15	=	=	SYM
m-904	96	1	(	(	PUNCT
m-904	96	2	𝑛1𝑥+𝑛0)𝑎1−(𝑚1𝑥+𝑚0)𝑏1	𝑛1𝑥+𝑛0)𝑎1−(𝑚1𝑥+𝑚0)𝑏1	NOUN
m-904	96	3	𝛼	𝛼	NOUN
m-904	96	4	=	=	SYM
m-904	96	5	(	(	PUNCT
m-904	96	6	𝑎1𝑛1−𝑏1𝑚1)𝑥+(𝑎1𝑛0−𝑏1𝑚0	𝑎1𝑛1−𝑏1𝑚1)𝑥+(𝑎1𝑛0−𝑏1𝑚0	NOUN
m-904	96	7	)	)	PUNCT
m-904	96	8	𝛼	𝛼	PROPN
m-904	96	9	(	(	PUNCT
m-904	96	10	32a	32a	NOUN
m-904	96	11	)	)	PUNCT
m-904	96	12	𝑦′(𝑥	𝑦′(𝑥	NOUN
m-904	96	13	)	)	PUNCT
m-904	96	14	=	=	SYM
m-904	96	15	(	(	PUNCT
m-904	96	16	𝑚1𝑥+𝑚0)𝑏0−(𝑛1𝑥+𝑛0)𝑎0	𝑚1𝑥+𝑚0)𝑏0−(𝑛1𝑥+𝑛0)𝑎0	PROPN
m-904	96	17	𝛼	𝛼	NOUN
m-904	96	18	=	=	SYM
m-904	96	19	(	(	PUNCT
m-904	96	20	𝑏0𝑚1−𝑎0𝑛1)𝑥+(𝑏0𝑚0−𝑎0𝑛0	𝑏0𝑚1−𝑎0𝑛1)𝑥+(𝑏0𝑚0−𝑎0𝑛0	NOUN
m-904	96	21	)	)	PUNCT
m-904	96	22	𝛼	𝛼	PROPN
m-904	96	23	(	(	PUNCT
m-904	96	24	32b	32b	NOUN
m-904	96	25	)	)	PUNCT
m-904	96	26	numerical	numerical	ADJ
m-904	96	27	example	example	NOUN
m-904	96	28	2	2	NUM
m-904	96	29	𝑦′	𝑦′	X
m-904	96	30	+	+	NOUN
m-904	96	31	3𝑦	3𝑦	NOUN
m-904	96	32	=	=	SYM
m-904	96	33	6𝑥	6𝑥	NOUN
m-904	96	34	+	+	CCONJ
m-904	96	35	2	2	NUM
m-904	96	36	−3𝑦′	−3𝑦′	PROPN
m-904	96	37	+	+	NUM
m-904	96	38	4𝑦	4𝑦	NUM
m-904	96	39	=	=	SYM
m-904	96	40	8𝑥	8𝑥	X
m-904	96	41	−	−	NUM
m-904	96	42	6	6	NUM
m-904	96	43	𝑎1	𝑎1	NOUN
m-904	96	44	=	=	SYM
m-904	96	45	1	1	NUM
m-904	96	46	,	,	PUNCT
m-904	96	47	𝑎0	𝑎0	PROPN
m-904	96	48	=	=	SYM
m-904	96	49	3	3	NUM
m-904	96	50	,	,	PUNCT
m-904	96	51	𝑏1	𝑏1	NOUN
m-904	96	52	=	=	SYM
m-904	96	53	−3	−3	ADJ
m-904	96	54	,	,	PUNCT
m-904	96	55	𝑏0	𝑏0	NOUN
m-904	96	56	=	=	SYM
m-904	96	57	4	4	NUM
m-904	96	58	,	,	PUNCT
m-904	96	59	𝑚1	𝑚1	NOUN
m-904	96	60	=	=	SYM
m-904	96	61	6	6	NUM
m-904	96	62	,	,	PUNCT
m-904	96	63	𝑚0	𝑚0	NOUN
m-904	96	64	=	=	SYM
m-904	96	65	2	2	NUM
m-904	96	66	,	,	PUNCT
m-904	96	67	𝑛1	𝑛1	ADJ
m-904	96	68	=	=	SYM
m-904	96	69	8	8	NUM
m-904	96	70	,	,	PUNCT
m-904	96	71	𝑛0	𝑛0	VERB
m-904	96	72	=	=	NOUN
m-904	96	73	−6	−6	NOUN
m-904	96	74	condition	condition	NOUN
m-904	96	75	1	1	NUM
m-904	96	76	:	:	PUNCT
m-904	96	77	𝑏0𝑚1	𝑏0𝑚1	NOUN
m-904	96	78	−	−	X
m-904	96	79	𝑎0𝑛1	𝑎0𝑛1	X
m-904	96	80	=	=	SYM
m-904	96	81	(	(	PUNCT
m-904	96	82	4)(6	4)(6	NUM
m-904	96	83	)	)	PUNCT
m-904	96	84	−	−	PROPN
m-904	96	85	(	(	PUNCT
m-904	96	86	3)(8	3)(8	NUM
m-904	96	87	)	)	PUNCT
m-904	96	88	=	=	SYM
m-904	96	89	0	0	PUNCT
m-904	97	1	=	=	ADJ
m-904	97	2	>	>	X
m-904	97	3	satisfied	satisfied	ADJ
m-904	97	4	condition	condition	NOUN
m-904	97	5	2	2	NUM
m-904	97	6	:	:	PUNCT
m-904	97	7	𝑎1𝑛1	𝑎1𝑛1	VERB
m-904	97	8	−	−	PROPN
m-904	98	1	𝑏1𝑚1	𝑏1𝑚1	X
m-904	98	2	=	=	PUNCT
m-904	98	3	𝑏0𝑚0	𝑏0𝑚0	PROPN
m-904	98	4	−	−	NOUN
m-904	98	5	𝑎0𝑛0	𝑎0𝑛0	PROPN
m-904	98	6	(	(	PUNCT
m-904	98	7	1)(8	1)(8	NUM
m-904	98	8	)	)	PUNCT
m-904	98	9	−	−	PROPN
m-904	98	10	(	(	PUNCT
m-904	98	11	−3)(6	−3)(6	X
m-904	98	12	)	)	PUNCT
m-904	98	13	=	=	SYM
m-904	98	14	(	(	PUNCT
m-904	98	15	4)(2	4)(2	NOUN
m-904	98	16	)	)	PUNCT
m-904	98	17	−	−	PROPN
m-904	98	18	(	(	PUNCT
m-904	98	19	3)(−6	3)(−6	NUM
m-904	98	20	)	)	PUNCT
m-904	98	21	8	8	NUM
m-904	99	1	+	+	SYM
m-904	99	2	18	18	NUM
m-904	99	3	=	=	SYM
m-904	99	4	8	8	NUM
m-904	99	5	+	+	NUM
m-904	99	6	18	18	NUM
m-904	99	7	=	=	NOUN
m-904	99	8	>	>	X
m-904	99	9	satisfied	satisfied	ADJ
m-904	99	10	𝛼	𝛼	NOUN
m-904	99	11	=	=	SYM
m-904	99	12	(	(	PUNCT
m-904	99	13	1)(4	1)(4	NUM
m-904	99	14	)	)	PUNCT
m-904	99	15	−	−	PROPN
m-904	99	16	(	(	PUNCT
m-904	99	17	3)(−3	3)(−3	NUM
m-904	99	18	)	)	PUNCT
m-904	100	1	=	=	SYM
m-904	100	2	13	13	NUM
m-904	100	3	𝑦(𝑥	𝑦(𝑥	NOUN
m-904	100	4	)	)	PUNCT
m-904	100	5	=	=	SYM
m-904	101	1	[	[	X
m-904	101	2	(	(	PUNCT
m-904	101	3	1)(8)−(−3)(6)]𝑥+[(1)(−6)−(−3)(2	1)(8)−(−3)(6)]𝑥+[(1)(−6)−(−3)(2	NUM
m-904	101	4	)	)	PUNCT
m-904	101	5	]	]	PUNCT
m-904	101	6	13	13	NUM
m-904	101	7	=	=	SYM
m-904	101	8	2𝑥	2𝑥	NUM
m-904	101	9	𝑦′(𝑥	𝑦′(𝑥	PROPN
m-904	101	10	)	)	PUNCT
m-904	101	11	=	=	PUNCT
m-904	102	1	[	[	X
m-904	102	2	(	(	PUNCT
m-904	102	3	4)(6)−(3)(8)]𝑥+[(4)(2)−(3)(−6	4)(6)−(3)(8)]𝑥+[(4)(2)−(3)(−6	NUM
m-904	102	4	)	)	PUNCT
m-904	102	5	]	]	PUNCT
m-904	102	6	13	13	NUM
m-904	102	7	=	=	SYM
m-904	102	8	2	2	NUM
m-904	102	9	check	check	NOUN
m-904	102	10	:	:	PUNCT
m-904	102	11	eq	eq	NOUN
m-904	102	12	1	1	NUM
m-904	102	13	:	:	PUNCT
m-904	102	14	𝑦′	𝑦′	X
m-904	102	15	+	+	NOUN
m-904	102	16	3𝑦	3𝑦	NOUN
m-904	102	17	=	=	SYM
m-904	102	18	2	2	NUM
m-904	102	19	+	+	SYM
m-904	102	20	3(2𝑥	3(2𝑥	NUM
m-904	102	21	)	)	PUNCT
m-904	102	22	=	=	NOUN
m-904	103	1	6𝑥	6𝑥	NOUN
m-904	103	2	+	+	CCONJ
m-904	103	3	2	2	NUM
m-904	103	4	√	√	PROPN
m-904	103	5	ijo	ijo	PROPN
m-904	103	6	international	international	PROPN
m-904	103	7	journal	journal	PROPN
m-904	103	8	of	of	ADP
m-904	103	9	mathematics	mathematics	PROPN
m-904	103	10	(	(	PUNCT
m-904	103	11	issn	issn	PROPN
m-904	103	12	:	:	PUNCT
m-904	103	13	2992	2992	NUM
m-904	103	14	-	-	SYM
m-904	103	15	4421	4421	NUM
m-904	103	16	)	)	PUNCT
m-904	103	17	ijo	ijo	PROPN
m-904	103	18	journals	journal	NOUN
m-904	103	19	volume	volume	NOUN
m-904	103	20	07	07	NUM
m-904	103	21	|	|	NOUN
m-904	103	22	issue	issue	NOUN
m-904	103	23	07	07	NUM
m-904	104	1	|	|	CCONJ
m-904	104	2	july	july	PROPN
m-904	104	3	2024	2024	NUM
m-904	104	4	|	|	ADV
m-904	104	5	https://ijojournals.com/index.php/m/index	https://ijojournals.com/index.php/m/index	NOUN
m-904	104	6	9	9	NUM
m-904	104	7	eq	eq	ADP
m-904	104	8	2	2	NUM
m-904	104	9	:	:	PUNCT
m-904	104	10	−3(2	−3(2	NOUN
m-904	104	11	)	)	PUNCT
m-904	105	1	+	+	CCONJ
m-904	106	1	4(2𝑥	4(2𝑥	NUM
m-904	106	2	)	)	PUNCT
m-904	106	3	=	=	VERB
m-904	106	4	8𝑥	8𝑥	X
m-904	106	5	−	−	NUM
m-904	107	1	6	6	NUM
m-904	107	2	√	√	NUM
m-904	107	3	4.3	4.3	NUM
m-904	107	4	case	case	NOUN
m-904	107	5	3	3	NUM
m-904	107	6	:	:	PUNCT
m-904	107	7	𝒇(𝒙	𝒇(𝒙	NUM
m-904	107	8	)	)	PUNCT
m-904	107	9	𝐚𝐧𝐝	𝐚𝐧𝐝	CCONJ
m-904	107	10	𝒈(𝒙	𝒈(𝒙	PROPN
m-904	107	11	)	)	PUNCT
m-904	107	12	are	be	AUX
m-904	107	13	exponential	exponential	ADJ
m-904	107	14	functions	function	NOUN
m-904	107	15	𝑓(𝑥	𝑓(𝑥	NOUN
m-904	107	16	)	)	PUNCT
m-904	107	17	=	=	SYM
m-904	107	18	𝑝𝑒𝑘𝑥	𝑝𝑒𝑘𝑥	NOUN
m-904	107	19	and	and	CCONJ
m-904	107	20	𝑔(𝑥	𝑔(𝑥	NUM
m-904	107	21	)	)	PUNCT
m-904	108	1	=	=	SYM
m-904	108	2	𝑞𝑒𝑘𝑥	𝑞𝑒𝑘𝑥	ADJ
m-904	108	3	,	,	PUNCT
m-904	108	4	𝑝	𝑝	PROPN
m-904	108	5	,	,	PUNCT
m-904	108	6	𝑞	𝑞	PROPN
m-904	108	7	,	,	PUNCT
m-904	108	8	𝑘	𝑘	X
m-904	108	9	=	=	PUNCT
m-904	108	10	constant	constant	ADJ
m-904	108	11	.	.	PUNCT
m-904	109	1	(	(	PUNCT
m-904	109	2	33	33	NUM
m-904	109	3	)	)	PUNCT
m-904	109	4	the	the	DET
m-904	109	5	necessary	necessary	ADJ
m-904	109	6	condition	condition	NOUN
m-904	109	7	is	be	AUX
m-904	109	8	𝑎1𝑞𝑘𝑒𝑘𝑥	𝑎1𝑞𝑘𝑒𝑘𝑥	PROPN
m-904	109	9	−	−	PROPN
m-904	109	10	𝑏1𝑝𝑘𝑒𝑘𝑥	𝑏1𝑝𝑘𝑒𝑘𝑥	PROPN
m-904	109	11	=	=	SYM
m-904	109	12	𝑏0𝑝𝑒𝑘𝑥	𝑏0𝑝𝑒𝑘𝑥	PROPN
m-904	109	13	−	−	NOUN
m-904	109	14	𝑎0𝑞𝑒𝑘𝑥	𝑎0𝑞𝑒𝑘𝑥	NOUN
m-904	109	15	(	(	PUNCT
m-904	109	16	34	34	NUM
m-904	109	17	)	)	PUNCT
m-904	109	18	which	which	PRON
m-904	109	19	simplifies	simplify	VERB
m-904	109	20	to	to	ADP
m-904	109	21	𝑘(𝑎1𝑞	𝑘(𝑎1𝑞	PROPN
m-904	109	22	−	−	PROPN
m-904	109	23	𝑏1𝑝	𝑏1𝑝	NUM
m-904	109	24	)	)	PUNCT
m-904	109	25	=	=	SYM
m-904	109	26	(	(	PUNCT
m-904	109	27	𝑏0𝑝	𝑏0𝑝	NOUN
m-904	109	28	−	−	PROPN
m-904	109	29	𝑎0𝑞	𝑎0𝑞	NOUN
m-904	109	30	)	)	PUNCT
m-904	109	31	.	.	PUNCT
m-904	110	1	(	(	PUNCT
m-904	110	2	35	35	NUM
m-904	110	3	)	)	PUNCT
m-904	110	4	using	use	VERB
m-904	110	5	the	the	DET
m-904	110	6	above	above	ADJ
m-904	110	7	theorem	theorem	NOUN
m-904	110	8	,	,	PUNCT
m-904	110	9	the	the	DET
m-904	110	10	solution	solution	NOUN
m-904	110	11	is	be	AUX
m-904	110	12	given	give	VERB
m-904	110	13	by	by	ADP
m-904	110	14	𝑦(𝑥	𝑦(𝑥	NOUN
m-904	110	15	)	)	PUNCT
m-904	111	1	=	=	SYM
m-904	111	2	𝑎1𝑞𝑒𝑘𝑥−𝑏1𝑝𝑒𝑘	𝑎1𝑞𝑒𝑘𝑥−𝑏1𝑝𝑒𝑘	NOUN
m-904	111	3	𝑎1𝑏0−𝑎0𝑏1	𝑎1𝑏0−𝑎0𝑏1	VERB
m-904	112	1	=	=	SYM
m-904	112	2	𝑎1𝑞−𝑏1𝑝	𝑎1𝑞−𝑏1𝑝	NOUN
m-904	112	3	𝛼	𝛼	PROPN
m-904	112	4	∙	∙	PROPN
m-904	112	5	𝑒𝑘𝑥	𝑒𝑘𝑥	X
m-904	112	6	(	(	PUNCT
m-904	112	7	36a	36a	NOUN
m-904	112	8	)	)	PUNCT
m-904	112	9	𝑦′(𝑥	𝑦′(𝑥	NOUN
m-904	112	10	)	)	PUNCT
m-904	112	11	=	=	SYM
m-904	112	12	𝑏0𝑝𝑒𝑘𝑥−𝑎0𝑞𝑒𝑘𝑥	𝑏0𝑝𝑒𝑘𝑥−𝑎0𝑞𝑒𝑘𝑥	NUM
m-904	112	13	𝑎1𝑏0−𝑎0𝑏1	𝑎1𝑏0−𝑎0𝑏1	NOUN
m-904	112	14	=	=	PUNCT
m-904	113	1	𝑏0𝑝−𝑎0𝑞	𝑏0𝑝−𝑎0𝑞	NOUN
m-904	113	2	𝛼	𝛼	PROPN
m-904	113	3	∙	∙	PROPN
m-904	113	4	𝑒𝑘𝑥	𝑒𝑘𝑥	X
m-904	113	5	(	(	PUNCT
m-904	113	6	36b	36b	NOUN
m-904	113	7	)	)	PUNCT
m-904	113	8	𝛼	𝛼	NOUN
m-904	113	9	=	=	PUNCT
m-904	113	10	𝑎1𝑏0	𝑎1𝑏0	PUNCT
m-904	113	11	−	−	PROPN
m-904	113	12	𝑎0𝑏1	𝑎0𝑏1	PUNCT
m-904	113	13	(	(	PUNCT
m-904	113	14	36c	36c	NOUN
m-904	113	15	)	)	PUNCT
m-904	113	16	numerical	numerical	ADJ
m-904	113	17	example	example	NOUN
m-904	113	18	3	3	NUM
m-904	113	19	2𝑦′	2𝑦′	NUM
m-904	114	1	+	+	NUM
m-904	114	2	3𝑦	3𝑦	NOUN
m-904	114	3	=	=	SYM
m-904	114	4	3𝑒3𝑥	3𝑒3𝑥	NOUN
m-904	114	5	𝑦′	𝑦′	X
m-904	114	6	−	−	PROPN
m-904	114	7	2𝑦	2𝑦	NOUN
m-904	114	8	=	=	SYM
m-904	114	9	1	1	NUM
m-904	114	10	3	3	NUM
m-904	114	11	𝑒3𝑥	𝑒3𝑥	NOUN
m-904	114	12	𝑎1	𝑎1	NOUN
m-904	114	13	=	=	SYM
m-904	114	14	2	2	NUM
m-904	114	15	,	,	PUNCT
m-904	114	16	𝑎0	𝑎0	PROPN
m-904	114	17	=	=	SYM
m-904	114	18	3	3	NUM
m-904	114	19	,	,	PUNCT
m-904	114	20	𝑏1	𝑏1	ADJ
m-904	114	21	=	=	SYM
m-904	114	22	1	1	NUM
m-904	114	23	,	,	PUNCT
m-904	114	24	𝑏0	𝑏0	NOUN
m-904	114	25	=	=	SYM
m-904	114	26	−2	−2	NOUN
m-904	114	27	,	,	PUNCT
m-904	114	28	𝑝	𝑝	NOUN
m-904	114	29	=	=	SYM
m-904	114	30	3	3	NUM
m-904	114	31	,	,	PUNCT
m-904	114	32	𝑞	𝑞	X
m-904	114	33	=	=	NOUN
m-904	114	34	1	1	NUM
m-904	114	35	3	3	NUM
m-904	114	36	,	,	PUNCT
m-904	114	37	𝑘	𝑘	X
m-904	114	38	=	=	SYM
m-904	114	39	3	3	NUM
m-904	114	40	𝑘(𝑎1𝑞	𝑘(𝑎1𝑞	ADV
m-904	114	41	−	−	PROPN
m-904	114	42	𝑏1𝑝	𝑏1𝑝	NUM
m-904	114	43	)	)	PUNCT
m-904	114	44	=	=	SYM
m-904	114	45	3	3	NUM
m-904	114	46	(	(	PUNCT
m-904	114	47	2	2	NUM
m-904	114	48	3	3	NUM
m-904	114	49	−	−	NOUN
m-904	114	50	3	3	NUM
m-904	114	51	)	)	PUNCT
m-904	114	52	=	=	PRON
m-904	114	53	−7	−7	NOUN
m-904	114	54	;	;	PUNCT
m-904	114	55	𝑏0𝑝	𝑏0𝑝	NOUN
m-904	114	56	−	−	NOUN
m-904	114	57	𝑎0𝑞	𝑎0𝑞	NUM
m-904	114	58	=	=	PUNCT
m-904	114	59	−6	−6	NOUN
m-904	115	1	−	−	PROPN
m-904	115	2	1	1	NUM
m-904	115	3	=	=	NOUN
m-904	115	4	−7	−7	NOUN
m-904	115	5	=	=	NOUN
m-904	115	6	>	>	X
m-904	115	7	condition	condition	NOUN
m-904	115	8	satisfied	satisfy	VERB
m-904	115	9	𝛼	𝛼	NOUN
m-904	115	10	=	=	X
m-904	115	11	(	(	PUNCT
m-904	115	12	2)(−2	2)(−2	NUM
m-904	115	13	)	)	PUNCT
m-904	115	14	−	−	PROPN
m-904	115	15	(	(	PUNCT
m-904	115	16	3)(1	3)(1	NUM
m-904	115	17	)	)	PUNCT
m-904	115	18	=	=	NOUN
m-904	115	19	−7	−7	NOUN
m-904	115	20	𝑦(𝑥	𝑦(𝑥	PROPN
m-904	115	21	)	)	PUNCT
m-904	115	22	=	=	SYM
m-904	115	23	(	(	PUNCT
m-904	115	24	2	2	NUM
m-904	115	25	)	)	PUNCT
m-904	115	26	(	(	PUNCT
m-904	115	27	1	1	NUM
m-904	115	28	3	3	NUM
m-904	115	29	)	)	PUNCT
m-904	115	30	−(1)(3	−(1)(3	NOUN
m-904	115	31	)	)	PUNCT
m-904	115	32	−7	−7	ADP
m-904	116	1	∙	∙	NOUN
m-904	116	2	𝑒3𝑥	𝑒3𝑥	PUNCT
m-904	116	3	=	=	SYM
m-904	116	4	1	1	NUM
m-904	116	5	3	3	NUM
m-904	116	6	∙	∙	PROPN
m-904	116	7	𝑒3𝑥	𝑒3𝑥	ADJ
m-904	116	8	𝑦′(𝑥	𝑦′(𝑥	NOUN
m-904	116	9	)	)	PUNCT
m-904	116	10	=	=	SYM
m-904	116	11	(	(	PUNCT
m-904	116	12	−2)(3)−(3	−2)(3)−(3	PROPN
m-904	116	13	)	)	PUNCT
m-904	116	14	(	(	PUNCT
m-904	116	15	1	1	NUM
m-904	116	16	3	3	NUM
m-904	116	17	)	)	PUNCT
m-904	116	18	−7	−7	ADP
m-904	116	19	∙	∙	NOUN
m-904	116	20	𝑒3𝑥	𝑒3𝑥	ADJ
m-904	116	21	=	=	SYM
m-904	116	22	𝑒3𝑥	𝑒3𝑥	ADJ
m-904	116	23	check	check	NOUN
m-904	116	24	:	:	PUNCT
m-904	116	25	eq	eq	NOUN
m-904	116	26	1	1	NUM
m-904	116	27	:	:	SYM
m-904	116	28	2𝑦′	2𝑦′	NUM
m-904	117	1	+	+	NUM
m-904	117	2	3𝑦	3𝑦	NOUN
m-904	117	3	=	=	SYM
m-904	117	4	2(𝑒3𝑥	2(𝑒3𝑥	NOUN
m-904	117	5	)	)	PUNCT
m-904	118	1	+	+	CCONJ
m-904	118	2	3	3	NUM
m-904	118	3	(	(	PUNCT
m-904	118	4	1	1	NUM
m-904	118	5	3	3	NUM
m-904	118	6	𝑒3𝑥	𝑒3𝑥	NOUN
m-904	118	7	)	)	PUNCT
m-904	118	8	=	=	SYM
m-904	119	1	3𝑒3𝑥	3𝑒3𝑥	NOUN
m-904	120	1	√	√	INTJ
m-904	120	2	eq	eq	ADP
m-904	120	3	2	2	NUM
m-904	120	4	:	:	PUNCT
m-904	120	5	𝑦′	𝑦′	X
m-904	120	6	−	−	PROPN
m-904	120	7	2𝑦	2𝑦	NOUN
m-904	120	8	=	=	SYM
m-904	120	9	(	(	PUNCT
m-904	120	10	𝑒3𝑥	𝑒3𝑥	NOUN
m-904	120	11	)	)	PUNCT
m-904	120	12	−	−	PROPN
m-904	120	13	2	2	NUM
m-904	120	14	(	(	PUNCT
m-904	120	15	1	1	NUM
m-904	120	16	3	3	NUM
m-904	120	17	𝑒3𝑥	𝑒3𝑥	NOUN
m-904	120	18	)	)	PUNCT
m-904	120	19	=	=	SYM
m-904	120	20	1	1	NUM
m-904	120	21	3	3	NUM
m-904	120	22	𝑒3𝑥	𝑒3𝑥	NOUN
m-904	120	23	√	√	PROPN
m-904	120	24	ijo	ijo	PROPN
m-904	120	25	international	international	PROPN
m-904	120	26	journal	journal	PROPN
m-904	120	27	of	of	ADP
m-904	120	28	mathematics	mathematics	PROPN
m-904	120	29	(	(	PUNCT
m-904	120	30	issn	issn	PROPN
m-904	120	31	:	:	PUNCT
m-904	120	32	2992	2992	NUM
m-904	120	33	-	-	SYM
m-904	120	34	4421	4421	NUM
m-904	120	35	)	)	PUNCT
m-904	120	36	ijo	ijo	PROPN
m-904	120	37	journals	journal	NOUN
m-904	120	38	volume	volume	NOUN
m-904	120	39	07	07	NUM
m-904	120	40	|	|	NOUN
m-904	120	41	issue	issue	NOUN
m-904	120	42	07	07	NUM
m-904	120	43	|	|	CCONJ
m-904	120	44	july	july	PROPN
m-904	120	45	2024	2024	NUM
m-904	121	1	|	|	ADV
m-904	121	2	https://ijojournals.com/index.php/m/index	https://ijojournals.com/index.php/m/index	NOUN
m-904	121	3	10	10	NUM
m-904	121	4	these	these	DET
m-904	121	5	calculations	calculation	NOUN
m-904	121	6	are	be	AUX
m-904	121	7	programmed	program	VERB
m-904	121	8	into	into	ADP
m-904	121	9	an	an	DET
m-904	121	10	excel	excel	NOUN
m-904	121	11	spreadsheet	spreadsheet	NOUN
m-904	121	12	as	as	SCONJ
m-904	121	13	shown	show	VERB
m-904	121	14	below	below	ADP
m-904	121	15	figure	figure	NOUN
m-904	121	16	2	2	NUM
m-904	121	17	.	.	PUNCT
m-904	121	18	one	one	NUM
m-904	121	19	just	just	ADV
m-904	121	20	have	have	VERB
m-904	121	21	to	to	PART
m-904	121	22	enter	enter	VERB
m-904	121	23	the	the	DET
m-904	121	24	corresponding	corresponding	ADJ
m-904	121	25	constants	constant	NOUN
m-904	121	26	and	and	CCONJ
m-904	121	27	coefficients	coefficient	NOUN
m-904	121	28	to	to	PART
m-904	121	29	obtain	obtain	VERB
m-904	121	30	the	the	DET
m-904	121	31	correct	correct	ADJ
m-904	121	32	answer	answer	NOUN
m-904	121	33	instantly	instantly	ADV
m-904	121	34	.	.	PUNCT
m-904	122	1	the	the	DET
m-904	122	2	necessary	necessary	ADJ
m-904	122	3	condition	condition	NOUN
m-904	122	4	is	be	AUX
m-904	122	5	also	also	ADV
m-904	122	6	checked	check	VERB
m-904	122	7	and	and	CCONJ
m-904	122	8	the	the	DET
m-904	122	9	solution	solution	NOUN
m-904	122	10	is	be	AUX
m-904	122	11	verified	verify	VERB
m-904	122	12	automatically	automatically	ADV
m-904	122	13	.	.	PUNCT
m-904	123	1	4.4	4.4	NUM
m-904	123	2	case4	case4	NOUN
m-904	123	3	:	:	PUNCT
m-904	124	1	𝒇(𝒙	𝒇(𝒙	NUM
m-904	124	2	)	)	PUNCT
m-904	124	3	𝐚𝐧𝐝	𝐚𝐧𝐝	CCONJ
m-904	124	4	𝒈(𝒙	𝒈(𝒙	PROPN
m-904	124	5	)	)	PUNCT
m-904	124	6	are	be	AUX
m-904	124	7	sine	sine	ADJ
m-904	124	8	and	and	CCONJ
m-904	124	9	cosine	cosine	NOUN
m-904	124	10	functions	function	NOUN
m-904	124	11	𝑓(𝑥	𝑓(𝑥	NOUN
m-904	124	12	)	)	PUNCT
m-904	124	13	=	=	SYM
m-904	125	1	𝑚	𝑚	ADP
m-904	125	2	sin	sin	NOUN
m-904	125	3	𝑥	𝑥	PROPN
m-904	125	4	+	+	CCONJ
m-904	125	5	𝑛	𝑛	VERB
m-904	125	6	cos	cos	ADP
m-904	125	7	𝑥	𝑥	PROPN
m-904	125	8	and	and	CCONJ
m-904	125	9	𝑔(𝑥	𝑔(𝑥	NUM
m-904	125	10	)	)	PUNCT
m-904	126	1	=	=	SYM
m-904	126	2	𝑝	𝑝	NOUN
m-904	126	3	sin	sin	NOUN
m-904	126	4	𝑥	𝑥	PROPN
m-904	126	5	+	+	CCONJ
m-904	126	6	𝑞	𝑞	X
m-904	126	7	cos	cos	ADP
m-904	126	8	𝑥	𝑥	PROPN
m-904	126	9	(	(	PUNCT
m-904	126	10	37	37	NUM
m-904	126	11	)	)	PUNCT
m-904	126	12	the	the	DET
m-904	126	13	necessary	necessary	ADJ
m-904	126	14	condition	condition	NOUN
m-904	126	15	is	be	AUX
m-904	126	16	𝑎1(𝑝	𝑎1(𝑝	PROPN
m-904	126	17	cos	cos	ADP
m-904	126	18	𝑥	𝑥	PROPN
m-904	126	19	−	−	NOUN
m-904	126	20	𝑞	𝑞	PART
m-904	126	21	sin	sin	VERB
m-904	126	22	𝑥	𝑥	NOUN
m-904	126	23	)	)	PUNCT
m-904	126	24	−	−	PROPN
m-904	127	1	𝑏1(𝑚	𝑏1(𝑚	PROPN
m-904	127	2	cos	cos	NOUN
m-904	127	3	𝑥	𝑥	PROPN
m-904	127	4	−	−	PROPN
m-904	127	5	𝑛	𝑛	PRON
m-904	127	6	sin	sin	NOUN
m-904	127	7	𝑥	𝑥	NOUN
m-904	127	8	)	)	PUNCT
m-904	127	9	=	=	SYM
m-904	127	10	𝑏0(𝑚	𝑏0(𝑚	PROPN
m-904	127	11	sin	sin	NOUN
m-904	127	12	𝑥	𝑥	NOUN
m-904	127	13	+	+	CCONJ
m-904	127	14	𝑛	𝑛	PROPN
m-904	127	15	cos	cos	NOUN
m-904	127	16	𝑥	𝑥	NOUN
m-904	127	17	)	)	PUNCT
m-904	127	18	−	−	PROPN
m-904	128	1	𝑎0(𝑝	𝑎0(𝑝	PRON
m-904	128	2	sin	sin	VERB
m-904	128	3	𝑥	𝑥	NOUN
m-904	128	4	+	+	CCONJ
m-904	128	5	𝑞	𝑞	X
m-904	128	6	cos	cos	PROPN
m-904	128	7	𝑥	𝑥	PROPN
m-904	128	8	)	)	PUNCT
m-904	128	9	sin	sin	NOUN
m-904	128	10	𝑥	𝑥	X
m-904	128	11	(	(	PUNCT
m-904	128	12	−𝑎1𝑞	−𝑎1𝑞	NUM
m-904	128	13	+	+	CCONJ
m-904	128	14	𝑏1𝑛	𝑏1𝑛	VERB
m-904	128	15	−	−	NOUN
m-904	128	16	𝑏0𝑚	𝑏0𝑚	PRON
m-904	128	17	+	+	CCONJ
m-904	128	18	𝑎0𝑝	𝑎0𝑝	PROPN
m-904	128	19	)	)	PUNCT
m-904	129	1	+	+	CCONJ
m-904	129	2	cos	cos	ADP
m-904	129	3	𝑥	𝑥	X
m-904	129	4	(	(	PUNCT
m-904	129	5	𝑎1𝑝	𝑎1𝑝	PROPN
m-904	129	6	−	−	PROPN
m-904	129	7	𝑏1𝑚	𝑏1𝑚	PROPN
m-904	129	8	−	−	PROPN
m-904	129	9	𝑏0𝑛	𝑏0𝑛	NOUN
m-904	129	10	+	+	CCONJ
m-904	129	11	𝑎0𝑞	𝑎0𝑞	NOUN
m-904	129	12	)	)	PUNCT
m-904	129	13	=	=	SYM
m-904	129	14	0	0	PUNCT
m-904	129	15	(	(	PUNCT
m-904	129	16	38	38	NUM
m-904	129	17	)	)	PUNCT
m-904	129	18	which	which	PRON
m-904	129	19	gives	give	VERB
m-904	129	20	𝑎1𝑞	𝑎1𝑞	PROPN
m-904	129	21	−	−	PROPN
m-904	129	22	𝑎0𝑝	𝑎0𝑝	PROPN
m-904	129	23	=	=	PUNCT
m-904	129	24	𝑏1𝑛	𝑏1𝑛	VERB
m-904	129	25	−	−	NOUN
m-904	129	26	𝑏0𝑚	𝑏0𝑚	NOUN
m-904	129	27	(	(	PUNCT
m-904	129	28	39a	39a	NUM
m-904	129	29	)	)	PUNCT
m-904	129	30	𝑎1𝑝	𝑎1𝑝	PROPN
m-904	129	31	+	+	NUM
m-904	129	32	𝑎0𝑞	𝑎0𝑞	PROPN
m-904	129	33	=	=	SYM
m-904	129	34	𝑏1𝑚	𝑏1𝑚	PROPN
m-904	130	1	+	+	CCONJ
m-904	130	2	𝑏0𝑛	𝑏0𝑛	SYM
m-904	130	3	(	(	PUNCT
m-904	130	4	39b	39b	NOUN
m-904	130	5	)	)	PUNCT
m-904	130	6	the	the	DET
m-904	130	7	solution	solution	NOUN
m-904	130	8	is	be	AUX
m-904	130	9	given	give	VERB
m-904	130	10	by	by	ADP
m-904	130	11	𝑦(𝑥	𝑦(𝑥	NOUN
m-904	130	12	)	)	PUNCT
m-904	130	13	=	=	PUNCT
m-904	130	14	(	(	PUNCT
m-904	130	15	𝑎1𝑝−𝑏1𝑚	𝑎1𝑝−𝑏1𝑚	NOUN
m-904	130	16	)	)	PUNCT
m-904	130	17	sin	sin	NOUN
m-904	130	18	𝑥+(𝑎1𝑞−𝑏1𝑛	𝑥+(𝑎1𝑞−𝑏1𝑛	NOUN
m-904	130	19	)	)	PUNCT
m-904	130	20	cos	cos	ADP
m-904	131	1	𝑥	𝑥	ADV
m-904	131	2	𝛼	𝛼	X
m-904	131	3	(	(	PUNCT
m-904	131	4	40a	40a	NUM
m-904	131	5	)	)	PUNCT
m-904	131	6	𝑦′(𝑥	𝑦′(𝑥	NOUN
m-904	131	7	)	)	PUNCT
m-904	131	8	=	=	SYM
m-904	131	9	(	(	PUNCT
m-904	131	10	𝑏0𝑚−𝑎0𝑝	𝑏0𝑚−𝑎0𝑝	NOUN
m-904	131	11	)	)	PUNCT
m-904	131	12	sin	sin	NOUN
m-904	131	13	𝑥+(𝑏0𝑛−𝑎0𝑞	𝑥+(𝑏0𝑛−𝑎0𝑞	NOUN
m-904	131	14	)	)	PUNCT
m-904	132	1	cos	cos	ADP
m-904	132	2	𝑥	𝑥	ADV
m-904	132	3	𝛼	𝛼	X
m-904	132	4	(	(	PUNCT
m-904	132	5	40b	40b	X
m-904	132	6	)	)	PUNCT
m-904	132	7	figure	figure	NOUN
m-904	132	8	2	2	NUM
m-904	132	9	:	:	PUNCT
m-904	132	10	spread	spread	VERB
m-904	132	11	sheet	sheet	NOUN
m-904	132	12	for	for	ADP
m-904	132	13	numerical	numerical	ADJ
m-904	132	14	example	example	PROPN
m-904	132	15	3	3	NUM
m-904	132	16	ijo	ijo	PROPN
m-904	132	17	international	international	PROPN
m-904	132	18	journal	journal	PROPN
m-904	132	19	of	of	ADP
m-904	132	20	mathematics	mathematics	PROPN
m-904	132	21	(	(	PUNCT
m-904	132	22	issn	issn	PROPN
m-904	132	23	:	:	PUNCT
m-904	132	24	2992	2992	NUM
m-904	132	25	-	-	SYM
m-904	132	26	4421	4421	NUM
m-904	132	27	)	)	PUNCT
m-904	132	28	ijo	ijo	PROPN
m-904	132	29	journals	journal	NOUN
m-904	132	30	volume	volume	NOUN
m-904	132	31	07	07	NUM
m-904	133	1	|	|	NOUN
m-904	133	2	issue	issue	NOUN
m-904	133	3	07	07	NUM
m-904	133	4	|	|	CCONJ
m-904	133	5	july	july	PROPN
m-904	133	6	2024	2024	NUM
m-904	134	1	|	|	ADV
m-904	134	2	https://ijojournals.com/index.php/m/index	https://ijojournals.com/index.php/m/index	NOUN
m-904	134	3	11	11	NUM
m-904	134	4	numerical	numerical	ADJ
m-904	134	5	example	example	NOUN
m-904	134	6	4	4	NUM
m-904	134	7	116𝑦′	116𝑦′	NUM
m-904	134	8	−	−	PROPN
m-904	134	9	58𝑦	58𝑦	NOUN
m-904	134	10	=	=	PUNCT
m-904	134	11	−7	−7	NOUN
m-904	134	12	sin	sin	VERB
m-904	134	13	𝑥	𝑥	PROPN
m-904	134	14	+	+	NUM
m-904	134	15	55	55	NUM
m-904	134	16	cos	cos	ADP
m-904	134	17	𝑥	𝑥	PROPN
m-904	134	18	35𝑦′	35𝑦′	NUM
m-904	134	19	+	+	NUM
m-904	134	20	15𝑦	15𝑦	NOUN
m-904	134	21	=	=	SYM
m-904	134	22	11	11	NUM
m-904	134	23	sin	sin	NOUN
m-904	134	24	𝑥	𝑥	PROPN
m-904	134	25	+	+	NUM
m-904	134	26	12	12	NUM
m-904	134	27	cos	cos	ADP
m-904	134	28	𝑥	𝑥	VERB
m-904	134	29	𝑎1	𝑎1	NOUN
m-904	134	30	=	=	SYM
m-904	134	31	116	116	NUM
m-904	134	32	,	,	PUNCT
m-904	134	33	𝑎0	𝑎0	PROPN
m-904	134	34	=	=	SYM
m-904	134	35	−58	−58	NOUN
m-904	134	36	,	,	PUNCT
m-904	134	37	𝑏1	𝑏1	ADJ
m-904	134	38	=	=	SYM
m-904	134	39	35	35	NUM
m-904	134	40	,	,	PUNCT
m-904	134	41	𝑏0	𝑏0	NOUN
m-904	134	42	=	=	SYM
m-904	134	43	15	15	NUM
m-904	134	44	,	,	PUNCT
m-904	134	45	𝑚	𝑚	NOUN
m-904	134	46	=	=	PUNCT
m-904	134	47	−7	−7	PROPN
m-904	134	48	,	,	PUNCT
m-904	134	49	𝑛	𝑛	PROPN
m-904	134	50	=	=	SYM
m-904	134	51	55	55	NUM
m-904	134	52	,	,	PUNCT
m-904	134	53	𝑝	𝑝	NOUN
m-904	134	54	=	=	SYM
m-904	134	55	11	11	NUM
m-904	134	56	,	,	PUNCT
m-904	134	57	𝑞	𝑞	X
m-904	134	58	=	=	PROPN
m-904	134	59	12	12	NUM
m-904	134	60	𝛼	𝛼	NOUN
m-904	134	61	=	=	PUNCT
m-904	134	62	𝑎1𝑏0	𝑎1𝑏0	PUNCT
m-904	134	63	−	−	VERB
m-904	134	64	𝑎0𝑏1	𝑎0𝑏1	PUNCT
m-904	134	65	=	=	SYM
m-904	134	66	3770	3770	NUM
m-904	134	67	necessary	necessary	ADJ
m-904	134	68	conditions	condition	NOUN
m-904	134	69	:	:	PUNCT
m-904	134	70	(	(	PUNCT
m-904	134	71	1	1	NUM
m-904	134	72	):	):	PUNCT
m-904	134	73	𝑎1𝑞	𝑎1𝑞	PROPN
m-904	134	74	−	−	PROPN
m-904	134	75	𝑎0𝑝	𝑎0𝑝	PROPN
m-904	134	76	=	=	PUNCT
m-904	134	77	𝑏1𝑛	𝑏1𝑛	VERB
m-904	134	78	−	−	NOUN
m-904	134	79	𝑏0𝑚	𝑏0𝑚	PRON
m-904	134	80	(	(	PUNCT
m-904	134	81	116)(12	116)(12	NOUN
m-904	134	82	)	)	PUNCT
m-904	134	83	−	−	PROPN
m-904	134	84	(	(	PUNCT
m-904	134	85	−58)(11	−58)(11	NOUN
m-904	134	86	)	)	PUNCT
m-904	134	87	=	=	SYM
m-904	134	88	(	(	PUNCT
m-904	134	89	35)(55	35)(55	NUM
m-904	134	90	)	)	PUNCT
m-904	134	91	−	−	PROPN
m-904	135	1	(	(	PUNCT
m-904	135	2	15)(−7	15)(−7	PROPN
m-904	135	3	)	)	PUNCT
m-904	135	4	2030	2030	NUM
m-904	135	5	=	=	SYM
m-904	135	6	2030	2030	NUM
m-904	136	1	=	=	SYM
m-904	136	2	>	>	X
m-904	136	3	condition	condition	NOUN
m-904	136	4	satisfied	satisfied	ADJ
m-904	136	5	(	(	PUNCT
m-904	136	6	2	2	NUM
m-904	136	7	):	):	PUNCT
m-904	136	8	𝑎1𝑝	𝑎1𝑝	PROPN
m-904	136	9	+	+	NUM
m-904	136	10	𝑎0𝑞	𝑎0𝑞	PROPN
m-904	136	11	=	=	SYM
m-904	136	12	𝑏1𝑚	𝑏1𝑚	PROPN
m-904	137	1	+	+	CCONJ
m-904	137	2	𝑏0𝑛	𝑏0𝑛	SYM
m-904	137	3	(	(	PUNCT
m-904	137	4	116)(11	116)(11	NOUN
m-904	137	5	)	)	PUNCT
m-904	137	6	+	+	CCONJ
m-904	137	7	(	(	PUNCT
m-904	137	8	−58)(12	−58)(12	ADP
m-904	137	9	)	)	PUNCT
m-904	137	10	=	=	SYM
m-904	137	11	(	(	PUNCT
m-904	137	12	35)(−7	35)(−7	NUM
m-904	137	13	)	)	PUNCT
m-904	137	14	+	+	CCONJ
m-904	137	15	(	(	PUNCT
m-904	137	16	15)(55	15)(55	NUM
m-904	137	17	)	)	PUNCT
m-904	137	18	580	580	NUM
m-904	137	19	=	=	SYM
m-904	137	20	580	580	NUM
m-904	137	21	=	=	NOUN
m-904	137	22	>	>	X
m-904	137	23	condition	condition	NOUN
m-904	137	24	satisfied	satisfy	VERB
m-904	137	25	𝑦(𝑥	𝑦(𝑥	NOUN
m-904	137	26	)	)	PUNCT
m-904	137	27	=	=	PUNCT
m-904	138	1	[	[	X
m-904	138	2	(	(	PUNCT
m-904	138	3	116)(11)−(35)(−7	116)(11)−(35)(−7	NOUN
m-904	138	4	)	)	PUNCT
m-904	138	5	]	]	PUNCT
m-904	138	6	sin	sin	NOUN
m-904	138	7	𝑥+[(116)(12)−(35)(55	𝑥+[(116)(12)−(35)(55	X
m-904	138	8	)	)	PUNCT
m-904	138	9	]	]	PUNCT
m-904	138	10	cos	cos	ADP
m-904	138	11	𝑥	𝑥	PROPN
m-904	138	12	3770	3770	NUM
m-904	138	13	=	=	SYM
m-904	138	14	0.403	0.403	NUM
m-904	138	15	sin	sin	VERB
m-904	138	16	𝑥	𝑥	PRON
m-904	138	17	−	−	NUM
m-904	138	18	0.141	0.141	NUM
m-904	138	19	cos	cos	ADP
m-904	138	20	𝑥	𝑥	PRON
m-904	138	21	𝑦′(𝑥	𝑦′(𝑥	PROPN
m-904	138	22	)	)	PUNCT
m-904	138	23	=	=	SYM
m-904	139	1	[	[	X
m-904	139	2	(	(	PUNCT
m-904	139	3	15)(−7)−(−58)(11	15)(−7)−(−58)(11	NUM
m-904	139	4	)	)	PUNCT
m-904	139	5	]	]	PUNCT
m-904	139	6	sin	sin	VERB
m-904	139	7	𝑥+[(15)(55)−(−58)(12	𝑥+[(15)(55)−(−58)(12	NUM
m-904	139	8	)	)	PUNCT
m-904	139	9	]	]	PUNCT
m-904	140	1	cos	cos	ADP
m-904	140	2	𝑥	𝑥	PROPN
m-904	140	3	3770	3770	NUM
m-904	140	4	=	=	SYM
m-904	140	5	0.141	0.141	NUM
m-904	140	6	sin	sin	NOUN
m-904	140	7	𝑥	𝑥	NOUN
m-904	140	8	+	+	NUM
m-904	140	9	0.403	0.403	NUM
m-904	140	10	cos	cos	ADP
m-904	140	11	𝑥	𝑥	PROPN
m-904	140	12	check	check	NOUN
m-904	140	13	:	:	PUNCT
m-904	140	14	eq	eq	NOUN
m-904	140	15	1	1	NUM
m-904	140	16	:	:	PUNCT
m-904	140	17	116𝑦′	116𝑦′	NUM
m-904	140	18	−	−	PROPN
m-904	140	19	58𝑦	58𝑦	NUM
m-904	140	20	=	=	SYM
m-904	140	21	(	(	PUNCT
m-904	140	22	16.356	16.356	NUM
m-904	140	23	sin	sin	NOUN
m-904	140	24	𝑥	𝑥	NOUN
m-904	140	25	+	+	CCONJ
m-904	140	26	46.748	46.748	NUM
m-904	140	27	cos	cos	ADJ
m-904	140	28	𝑥	𝑥	NOUN
m-904	140	29	)	)	PUNCT
m-904	140	30	−	−	PROPN
m-904	140	31	(	(	PUNCT
m-904	140	32	23.374	23.374	NUM
m-904	140	33	sin	sin	NOUN
m-904	140	34	𝑥	𝑥	PRON
m-904	140	35	−	−	PROPN
m-904	140	36	8.178	8.178	NUM
m-904	140	37	cos	cos	NOUN
m-904	140	38	𝑥	𝑥	NOUN
m-904	140	39	)	)	PUNCT
m-904	140	40	=	=	SYM
m-904	140	41	−7.018	−7.018	NOUN
m-904	140	42	sin	sin	VERB
m-904	140	43	𝑥	𝑥	PROPN
m-904	140	44	+	+	NUM
m-904	140	45	54.926	54.926	NUM
m-904	140	46	cos	cos	NOUN
m-904	140	47	𝑥	𝑥	NOUN
m-904	140	48	√	√	NUM
m-904	140	49	eq	eq	ADP
m-904	140	50	2	2	NUM
m-904	140	51	:	:	SYM
m-904	140	52	35𝑦′	35𝑦′	NUM
m-904	140	53	+	+	NUM
m-904	140	54	15𝑦	15𝑦	NOUN
m-904	140	55	=	=	SYM
m-904	140	56	(	(	PUNCT
m-904	140	57	4.935	4.935	NUM
m-904	140	58	sin	sin	NOUN
m-904	140	59	𝑥	𝑥	PROPN
m-904	140	60	+	+	NUM
m-904	140	61	14.105	14.105	NUM
m-904	140	62	cos	cos	NOUN
m-904	140	63	𝑥	𝑥	NOUN
m-904	140	64	)	)	PUNCT
m-904	140	65	+	+	CCONJ
m-904	140	66	(	(	PUNCT
m-904	140	67	6.045	6.045	NUM
m-904	140	68	sin	sin	NOUN
m-904	140	69	𝑥	𝑥	PRON
m-904	140	70	−	−	PROPN
m-904	140	71	2.115	2.115	NUM
m-904	140	72	cos	cos	NOUN
m-904	140	73	𝑥	𝑥	NOUN
m-904	140	74	)	)	PUNCT
m-904	140	75	=	=	SYM
m-904	140	76	10.980	10.980	NUM
m-904	140	77	sin	sin	NOUN
m-904	140	78	𝑥	𝑥	X
m-904	141	1	+	+	NUM
m-904	141	2	11.990	11.990	NUM
m-904	141	3	cos	cos	NOUN
m-904	141	4	𝑥	𝑥	PROPN
m-904	141	5	√	√	NUM
m-904	141	6	5	5	NUM
m-904	141	7	.	.	PUNCT
m-904	141	8	conclusion	conclusion	NOUN
m-904	141	9	a	a	DET
m-904	141	10	set	set	NOUN
m-904	141	11	of	of	ADP
m-904	141	12	two	two	NUM
m-904	141	13	non	non	ADJ
m-904	141	14	-	-	ADJ
m-904	141	15	homogeneous	homogeneous	ADJ
m-904	141	16	linear	linear	ADJ
m-904	141	17	first	first	ADJ
m-904	141	18	-	-	PUNCT
m-904	141	19	order	order	NOUN
m-904	141	20	simultaneous	simultaneous	ADJ
m-904	141	21	ordinary	ordinary	ADJ
m-904	141	22	differential	differential	ADJ
m-904	141	23	equations	equation	NOUN
m-904	141	24	with	with	ADP
m-904	141	25	constant	constant	ADJ
m-904	141	26	coefficients	coefficient	NOUN
m-904	141	27	of	of	ADP
m-904	141	28	a	a	DET
m-904	141	29	certain	certain	ADJ
m-904	141	30	form	form	NOUN
m-904	141	31	are	be	AUX
m-904	141	32	studied	study	VERB
m-904	141	33	.	.	PUNCT
m-904	142	1	four	four	NUM
m-904	142	2	different	different	ADJ
m-904	142	3	types	type	NOUN
m-904	142	4	of	of	ADP
m-904	142	5	the	the	DET
m-904	142	6	forcing	force	VERB
m-904	142	7	functions	function	NOUN
m-904	142	8	,	,	PUNCT
m-904	142	9	namely	namely	ADV
m-904	142	10	constants	constant	NOUN
m-904	142	11	,	,	PUNCT
m-904	142	12	linear	linear	ADJ
m-904	142	13	functions	function	NOUN
m-904	142	14	,	,	PUNCT
m-904	142	15	natural	natural	ADJ
m-904	142	16	exponential	exponential	ADJ
m-904	142	17	functions	function	NOUN
m-904	142	18	and	and	CCONJ
m-904	142	19	sinusoidal	sinusoidal	NOUN
m-904	142	20	functions	function	NOUN
m-904	142	21	,	,	PUNCT
m-904	142	22	have	have	AUX
m-904	142	23	been	be	AUX
m-904	142	24	investigated	investigate	VERB
m-904	142	25	.	.	PUNCT
m-904	143	1	for	for	ADP
m-904	143	2	each	each	DET
m-904	143	3	type	type	NOUN
m-904	143	4	,	,	PUNCT
m-904	143	5	an	an	DET
m-904	143	6	algebraic	algebraic	ADJ
m-904	143	7	equation	equation	NOUN
m-904	143	8	to	to	PART
m-904	143	9	determine	determine	VERB
m-904	143	10	the	the	DET
m-904	143	11	solution	solution	NOUN
m-904	143	12	and	and	CCONJ
m-904	143	13	its	its	PRON
m-904	143	14	derivative	derivative	NOUN
m-904	143	15	as	as	ADV
m-904	143	16	well	well	ADV
m-904	143	17	as	as	ADP
m-904	143	18	the	the	DET
m-904	143	19	required	require	VERB
m-904	143	20	necessary	necessary	ADJ
m-904	143	21	conditions	condition	NOUN
m-904	143	22	are	be	AUX
m-904	143	23	derived	derive	VERB
m-904	143	24	.	.	PUNCT
m-904	144	1	moreover	moreover	ADV
m-904	144	2	,	,	PUNCT
m-904	144	3	these	these	DET
m-904	144	4	simple	simple	ADJ
m-904	144	5	algebraic	algebraic	ADJ
m-904	144	6	formulae	formulae	NOUN
m-904	144	7	can	can	AUX
m-904	144	8	be	be	AUX
m-904	144	9	easily	easily	ADV
m-904	144	10	programmed	program	VERB
m-904	144	11	into	into	ADP
m-904	144	12	a	a	DET
m-904	144	13	spreadsheet	spreadsheet	NOUN
m-904	144	14	which	which	PRON
m-904	144	15	can	can	AUX
m-904	144	16	automatically	automatically	ADV
m-904	144	17	compute	compute	VERB
m-904	144	18	the	the	DET
m-904	144	19	solution	solution	NOUN
m-904	144	20	,	,	PUNCT
m-904	144	21	check	check	VERB
m-904	144	22	the	the	DET
m-904	144	23	necessary	necessary	ADJ
m-904	144	24	condition	condition	NOUN
m-904	144	25	and	and	CCONJ
m-904	144	26	verify	verify	VERB
m-904	144	27	the	the	DET
m-904	144	28	answer	answer	NOUN
m-904	144	29	.	.	PUNCT
m-904	145	1	ijo	ijo	PROPN
m-904	145	2	international	international	PROPN
m-904	145	3	journal	journal	PROPN
m-904	145	4	of	of	ADP
m-904	145	5	mathematics	mathematics	PROPN
m-904	145	6	(	(	PUNCT
m-904	145	7	issn	issn	PROPN
m-904	145	8	:	:	PUNCT
m-904	145	9	2992	2992	NUM
m-904	145	10	-	-	SYM
m-904	145	11	4421	4421	NUM
m-904	145	12	)	)	PUNCT
m-904	145	13	ijo	ijo	PROPN
m-904	145	14	journals	journal	NOUN
m-904	145	15	volume	volume	NOUN
m-904	145	16	07	07	NUM
m-904	145	17	|	|	NOUN
m-904	145	18	issue	issue	NOUN
m-904	145	19	07	07	NUM
m-904	146	1	|	|	CCONJ
m-904	146	2	july	july	PROPN
m-904	146	3	2024	2024	NUM
m-904	146	4	|	|	ADV
m-904	146	5	https://ijojournals.com/index.php/m/index	https://ijojournals.com/index.php/m/index	NOUN
m-904	146	6	12	12	NUM
m-904	146	7	6	6	NUM
m-904	146	8	.	.	PUNCT
m-904	147	1	discussion	discussion	NOUN
m-904	147	2	the	the	DET
m-904	147	3	theorem	theorem	NOUN
m-904	147	4	and	and	CCONJ
m-904	147	5	formulae	formulae	ADJ
m-904	147	6	derived	derive	VERB
m-904	147	7	are	be	AUX
m-904	147	8	applicable	applicable	ADJ
m-904	147	9	only	only	ADV
m-904	147	10	to	to	ADP
m-904	147	11	one	one	NUM
m-904	147	12	special	special	ADJ
m-904	147	13	form	form	NOUN
m-904	147	14	of	of	ADP
m-904	147	15	nonhomogeneous	nonhomogeneous	ADJ
m-904	147	16	linear	linear	ADJ
m-904	147	17	first	first	ADJ
m-904	147	18	-	-	PUNCT
m-904	147	19	order	order	NOUN
m-904	147	20	simultaneous	simultaneous	ADJ
m-904	147	21	ordinary	ordinary	ADJ
m-904	147	22	differential	differential	ADJ
m-904	147	23	equations	equation	NOUN
m-904	147	24	with	with	ADP
m-904	147	25	constant	constant	ADJ
m-904	147	26	coefficients	coefficient	NOUN
m-904	147	27	.	.	PUNCT
m-904	148	1	this	this	PRON
m-904	148	2	is	be	AUX
m-904	148	3	indeed	indeed	ADV
m-904	148	4	restrictive	restrictive	ADJ
m-904	148	5	.	.	PUNCT
m-904	149	1	moreover	moreover	ADV
m-904	149	2	,	,	PUNCT
m-904	149	3	this	this	DET
m-904	149	4	method	method	NOUN
m-904	149	5	only	only	ADV
m-904	149	6	works	work	VERB
m-904	149	7	for	for	ADP
m-904	149	8	the	the	DET
m-904	149	9	stated	state	VERB
m-904	149	10	four	four	NUM
m-904	149	11	cases	case	NOUN
m-904	149	12	of	of	ADP
m-904	149	13	the	the	DET
m-904	149	14	forcing	force	VERB
m-904	149	15	function	function	NOUN
m-904	149	16	and	and	CCONJ
m-904	149	17	a	a	DET
m-904	149	18	stringent	stringent	ADJ
m-904	149	19	necessary	necessary	ADJ
m-904	149	20	condition	condition	NOUN
m-904	149	21	must	must	AUX
m-904	149	22	be	be	AUX
m-904	149	23	satisfied	satisfied	ADJ
m-904	149	24	.	.	PUNCT
m-904	150	1	however	however	ADV
m-904	150	2	,	,	PUNCT
m-904	150	3	most	most	ADJ
m-904	150	4	of	of	ADP
m-904	150	5	the	the	DET
m-904	150	6	existing	exist	VERB
m-904	150	7	methods	method	NOUN
m-904	150	8	have	have	VERB
m-904	150	9	their	their	PRON
m-904	150	10	restrictions	restriction	NOUN
m-904	150	11	too	too	ADV
m-904	150	12	and	and	CCONJ
m-904	150	13	can	can	AUX
m-904	150	14	not	not	PART
m-904	150	15	be	be	AUX
m-904	150	16	used	use	VERB
m-904	150	17	to	to	PART
m-904	150	18	solve	solve	VERB
m-904	150	19	any	any	DET
m-904	150	20	form	form	NOUN
m-904	150	21	of	of	ADP
m-904	150	22	simultaneous	simultaneous	ADJ
m-904	150	23	differential	differential	ADJ
m-904	150	24	equations	equation	NOUN
m-904	150	25	with	with	ADP
m-904	150	26	different	different	ADJ
m-904	150	27	combinations	combination	NOUN
m-904	150	28	of	of	ADP
m-904	150	29	the	the	DET
m-904	150	30	independent	independent	ADJ
m-904	150	31	and	and	CCONJ
m-904	150	32	dependent	dependent	ADJ
m-904	150	33	variables	variable	NOUN
m-904	150	34	and	and	CCONJ
m-904	150	35	their	their	PRON
m-904	150	36	derivatives	derivative	NOUN
m-904	150	37	.	.	PUNCT
m-904	151	1	despite	despite	SCONJ
m-904	151	2	its	its	PRON
m-904	151	3	limitations	limitation	NOUN
m-904	151	4	,	,	PUNCT
m-904	151	5	the	the	DET
m-904	151	6	proposed	propose	VERB
m-904	151	7	theorem	theorem	NOUN
m-904	151	8	and	and	CCONJ
m-904	151	9	the	the	DET
m-904	151	10	resulting	result	VERB
m-904	151	11	formulae	formulae	NOUN
m-904	151	12	have	have	VERB
m-904	151	13	the	the	DET
m-904	151	14	advantage	advantage	NOUN
m-904	151	15	that	that	SCONJ
m-904	151	16	they	they	PRON
m-904	151	17	are	be	AUX
m-904	151	18	algebraic	algebraic	ADJ
m-904	151	19	and	and	CCONJ
m-904	151	20	can	can	AUX
m-904	151	21	be	be	AUX
m-904	151	22	easily	easily	ADV
m-904	151	23	programmed	program	VERB
m-904	151	24	into	into	ADP
m-904	151	25	a	a	DET
m-904	151	26	spreadsheet	spreadsheet	NOUN
m-904	151	27	that	that	PRON
m-904	151	28	can	can	AUX
m-904	151	29	perform	perform	VERB
m-904	151	30	repeated	repeat	VERB
m-904	151	31	computations	computation	NOUN
m-904	151	32	with	with	ADP
m-904	151	33	different	different	ADJ
m-904	151	34	constants	constant	NOUN
m-904	151	35	and	and	CCONJ
m-904	151	36	coefficients	coefficient	NOUN
m-904	151	37	effortlessly	effortlessly	ADV
m-904	151	38	.	.	PUNCT
m-904	152	1	the	the	DET
m-904	152	2	spreadsheet	spreadsheet	NOUN
m-904	152	3	can	can	AUX
m-904	152	4	also	also	ADV
m-904	152	5	be	be	AUX
m-904	152	6	programmed	program	VERB
m-904	152	7	to	to	PART
m-904	152	8	check	check	VERB
m-904	152	9	the	the	DET
m-904	152	10	necessary	necessary	ADJ
m-904	152	11	condition	condition	NOUN
m-904	152	12	and	and	CCONJ
m-904	152	13	verify	verify	VERB
m-904	152	14	the	the	DET
m-904	152	15	solution	solution	NOUN
m-904	152	16	.	.	PUNCT
m-904	153	1	the	the	DET
m-904	153	2	investigation	investigation	NOUN
m-904	153	3	employing	employ	VERB
m-904	153	4	this	this	DET
m-904	153	5	technique	technique	NOUN
m-904	153	6	to	to	ADP
m-904	153	7	different	different	ADJ
m-904	153	8	forms	form	NOUN
m-904	153	9	of	of	ADP
m-904	153	10	simultaneous	simultaneous	ADJ
m-904	153	11	ode	ode	NOUN
m-904	153	12	and	and	CCONJ
m-904	153	13	with	with	ADP
m-904	153	14	different	different	ADJ
m-904	153	15	forcing	force	VERB
m-904	153	16	functions	function	NOUN
m-904	153	17	would	would	AUX
m-904	153	18	be	be	AUX
m-904	153	19	an	an	DET
m-904	153	20	interesting	interesting	ADJ
m-904	153	21	endeavor	endeavor	NOUN
m-904	153	22	.	.	PUNCT
m-904	154	1	ijo	ijo	PROPN
m-904	154	2	international	international	PROPN
m-904	154	3	journal	journal	PROPN
m-904	154	4	of	of	ADP
m-904	154	5	mathematics	mathematics	PROPN
m-904	154	6	(	(	PUNCT
m-904	154	7	issn	issn	PROPN
m-904	154	8	:	:	PUNCT
m-904	154	9	2992	2992	NUM
m-904	154	10	-	-	SYM
m-904	154	11	4421	4421	NUM
m-904	154	12	)	)	PUNCT
m-904	154	13	ijo	ijo	PROPN
m-904	154	14	journals	journal	NOUN
m-904	154	15	volume	volume	NOUN
m-904	154	16	07	07	NUM
m-904	154	17	|	|	NOUN
m-904	154	18	issue	issue	NOUN
m-904	154	19	07	07	NUM
m-904	155	1	|	|	CCONJ
m-904	155	2	july	july	PROPN
m-904	155	3	2024	2024	NUM
m-904	155	4	|	|	ADV
m-904	155	5	https://ijojournals.com/index.php/m/index	https://ijojournals.com/index.php/m/index	NOUN
m-904	155	6	13	13	NUM
m-904	155	7	references	reference	NOUN
m-904	155	8	1	1	NUM
m-904	155	9	.	.	PUNCT
m-904	155	10	andrilli	andrilli	PROPN
m-904	155	11	,	,	PUNCT
m-904	155	12	s.	s.	PROPN
m-904	155	13	&	&	CCONJ
m-904	155	14	hecker	hecker	PROPN
m-904	155	15	,	,	PUNCT
m-904	155	16	d.	d.	PROPN
m-904	155	17	(	(	PUNCT
m-904	155	18	2019	2019	NUM
m-904	155	19	)	)	PUNCT
m-904	155	20	.	.	PUNCT
m-904	156	1	elementary	elementary	PROPN
m-904	156	2	linear	linear	PROPN
m-904	156	3	algebra	algebra	PROPN
m-904	156	4	(	(	PUNCT
m-904	156	5	5th	5th	ADJ
m-904	156	6	edn	edn	NOUN
m-904	156	7	)	)	PUNCT
m-904	156	8	.	.	PUNCT
m-904	157	1	harcourt	harcourt	PROPN
m-904	157	2	academic	academic	PROPN
m-904	157	3	press	press	PROPN
m-904	157	4	,	,	PUNCT
m-904	157	5	san	san	PROPN
m-904	157	6	diego	diego	PROPN
m-904	157	7	.	.	PUNCT
m-904	158	1	2	2	X
m-904	158	2	.	.	X
m-904	158	3	kolman	kolman	PROPN
m-904	158	4	,	,	PUNCT
m-904	158	5	b.	b.	PROPN
m-904	158	6	&	&	CCONJ
m-904	158	7	hill	hill	PROPN
m-904	158	8	,	,	PUNCT
m-904	158	9	d.	d.	PROPN
m-904	158	10	(	(	PUNCT
m-904	158	11	2007	2007	NUM
m-904	158	12	)	)	PUNCT
m-904	158	13	.	.	PUNCT
m-904	159	1	elementary	elementary	PROPN
m-904	159	2	linear	linear	PROPN
m-904	159	3	algebra	algebra	PROPN
m-904	159	4	(	(	PUNCT
m-904	159	5	9th	9th	ADJ
m-904	159	6	edn	edn	NOUN
m-904	159	7	)	)	PUNCT
m-904	159	8	.	.	PUNCT
m-904	160	1	macmillan	macmillan	PROPN
m-904	160	2	publishing	publishing	PROPN
m-904	160	3	company	company	NOUN
m-904	160	4	,	,	PUNCT
m-904	160	5	new	new	PROPN
m-904	160	6	york	york	PROPN
m-904	160	7	.	.	PUNCT
m-904	161	1	3	3	X
m-904	161	2	.	.	X
m-904	161	3	boyce	boyce	PROPN
m-904	161	4	,	,	PUNCT
m-904	161	5	w.e	w.e	PROPN
m-904	161	6	.	.	PROPN
m-904	161	7	(	(	PUNCT
m-904	161	8	2017	2017	NUM
m-904	161	9	)	)	PUNCT
m-904	161	10	.	.	PUNCT
m-904	162	1	elementary	elementary	ADJ
m-904	162	2	differential	differential	PROPN
m-904	162	3	equations	equation	NOUN
m-904	162	4	and	and	CCONJ
m-904	162	5	boundary	boundary	ADJ
m-904	162	6	value	value	NOUN
m-904	162	7	problems	problem	NOUN
m-904	162	8	(	(	PUNCT
m-904	162	9	11th	11th	ADJ
m-904	162	10	edn	edn	NOUN
m-904	162	11	)	)	PUNCT
m-904	162	12	.	.	PUNCT
m-904	163	1	wiley	wiley	PROPN
m-904	163	2	.	.	PUNCT
m-904	164	1	4	4	X
m-904	164	2	.	.	X
m-904	164	3	edwards	edwards	PROPN
m-904	164	4	,	,	PUNCT
m-904	164	5	h.	h.	PROPN
m-904	164	6	&	&	CCONJ
m-904	164	7	penney	penney	PROPN
m-904	164	8	,	,	PUNCT
m-904	164	9	d.	d.	PROPN
m-904	164	10	(	(	PUNCT
m-904	164	11	2019	2019	NUM
m-904	164	12	)	)	PUNCT
m-904	164	13	.	.	PUNCT
m-904	165	1	elementary	elementary	ADJ
m-904	165	2	differential	differential	PROPN
m-904	165	3	equations	equation	NOUN
m-904	165	4	with	with	ADP
m-904	165	5	boundary	boundary	ADJ
m-904	165	6	value	value	NOUN
m-904	165	7	problems	problem	NOUN
m-904	165	8	(	(	PUNCT
m-904	165	9	6th	6th	ADJ
m-904	165	10	edn	edn	NOUN
m-904	165	11	)	)	PUNCT
m-904	165	12	.	.	PUNCT
m-904	166	1	pearson	pearson	PROPN
m-904	166	2	.	.	PUNCT
m-904	167	1	5	5	NUM
m-904	167	2	.	.	X
m-904	167	3	kohler	kohler	NOUN
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m-904	182	7	|	|	ADV
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