id	sid	tid	token	lemma	pos
ajrhss-1265	1	1	american	american	PROPN
ajrhss-1265	1	2	journal	journal	PROPN
ajrhss-1265	1	3	of	of	ADP
ajrhss-1265	1	4	research	research	NOUN
ajrhss-1265	1	5	in	in	ADP
ajrhss-1265	1	6	humanities	humanity	NOUN
ajrhss-1265	1	7	and	and	CCONJ
ajrhss-1265	1	8	social	social	ADJ
ajrhss-1265	1	9	sciences	science	NOUN
ajrhss-1265	1	10	issn	issn	PROPN
ajrhss-1265	1	11	(	(	PUNCT
ajrhss-1265	1	12	e	e	NOUN
ajrhss-1265	1	13	):	):	PUNCT
ajrhss-1265	1	14	2832	2832	NUM
ajrhss-1265	1	15	-	-	SYM
ajrhss-1265	1	16	8019	8019	NUM
ajrhss-1265	1	17	volume	volume	NOUN
ajrhss-1265	1	18	16	16	NUM
ajrhss-1265	1	19	,	,	PUNCT
ajrhss-1265	1	20	|	|	ADV
ajrhss-1265	1	21	sep	sep	PROPN
ajrhss-1265	1	22	.	.	PROPN
ajrhss-1265	1	23	,	,	PUNCT
ajrhss-1265	1	24	2023	2023	NUM
ajrhss-1265	1	25	p	p	NOUN
ajrhss-1265	1	26	a	a	DET
ajrhss-1265	1	27	g	g	NOUN
ajrhss-1265	1	28	e	e	NOUN
ajrhss-1265	1	29	|	|	NOUN
ajrhss-1265	1	30	88	88	NUM
ajrhss-1265	1	31	www.americanjournal.org	www.americanjournal.org	NOUN
ajrhss-1265	1	32	basics	basic	NOUN
ajrhss-1265	1	33	of	of	ADP
ajrhss-1265	1	34	differential	differential	ADJ
ajrhss-1265	1	35	equations	equation	NOUN
ajrhss-1265	1	36	kazakova	kazakova	PROPN
ajrhss-1265	1	37	madina	madina	PROPN
ajrhss-1265	1	38	allayar	allayar	PROPN
ajrhss-1265	1	39	qizi	qizi	PROPN
ajrhss-1265	1	40	a	a	DET
ajrhss-1265	1	41	b	b	PROPN
ajrhss-1265	1	42	s	s	ADP
ajrhss-1265	1	43	t	t	PROPN
ajrhss-1265	1	44	r	r	NOUN
ajrhss-1265	1	45	a	a	DET
ajrhss-1265	1	46	c	c	NOUN
ajrhss-1265	1	47	t	t	NOUN
ajrhss-1265	1	48	k	k	X
ajrhss-1265	1	49	e	e	PROPN
ajrhss-1265	1	50	y	y	PROPN
ajrhss-1265	1	51	w	w	NOUN
ajrhss-1265	1	52	o	o	NOUN
ajrhss-1265	1	53	r	r	NOUN
ajrhss-1265	1	54	d	d	NOUN
ajrhss-1265	1	55	s	s	PART
ajrhss-1265	1	56	calculus	calculus	NOUN
ajrhss-1265	1	57	is	be	AUX
ajrhss-1265	1	58	the	the	DET
ajrhss-1265	1	59	mathematics	mathematic	NOUN
ajrhss-1265	1	60	of	of	ADP
ajrhss-1265	1	61	change	change	NOUN
ajrhss-1265	1	62	,	,	PUNCT
ajrhss-1265	1	63	and	and	CCONJ
ajrhss-1265	1	64	rates	rate	NOUN
ajrhss-1265	1	65	of	of	ADP
ajrhss-1265	1	66	change	change	NOUN
ajrhss-1265	1	67	are	be	AUX
ajrhss-1265	1	68	expressed	express	VERB
ajrhss-1265	1	69	by	by	ADP
ajrhss-1265	1	70	derivatives	derivative	NOUN
ajrhss-1265	1	71	.	.	PUNCT
ajrhss-1265	2	1	thus	thus	ADV
ajrhss-1265	2	2	,	,	PUNCT
ajrhss-1265	2	3	one	one	NUM
ajrhss-1265	2	4	of	of	ADP
ajrhss-1265	2	5	the	the	DET
ajrhss-1265	2	6	most	most	ADV
ajrhss-1265	2	7	common	common	ADJ
ajrhss-1265	2	8	ways	way	NOUN
ajrhss-1265	2	9	to	to	PART
ajrhss-1265	2	10	use	use	VERB
ajrhss-1265	2	11	calculus	calculus	NOUN
ajrhss-1265	2	12	is	be	AUX
ajrhss-1265	2	13	to	to	PART
ajrhss-1265	2	14	set	set	VERB
ajrhss-1265	2	15	up	up	ADP
ajrhss-1265	2	16	an	an	DET
ajrhss-1265	2	17	equation	equation	NOUN
ajrhss-1265	2	18	containing	contain	VERB
ajrhss-1265	2	19	an	an	DET
ajrhss-1265	2	20	unknown	unknown	ADJ
ajrhss-1265	2	21	function	function	NOUN
ajrhss-1265	2	22	y	y	PROPN
ajrhss-1265	2	23	=	=	PROPN
ajrhss-1265	2	24	f(x	f(x	PROPN
ajrhss-1265	2	25	)	)	PUNCT
ajrhss-1265	2	26	and	and	CCONJ
ajrhss-1265	2	27	its	its	PRON
ajrhss-1265	2	28	derivative	derivative	NOUN
ajrhss-1265	2	29	,	,	PUNCT
ajrhss-1265	2	30	known	know	VERB
ajrhss-1265	2	31	as	as	ADP
ajrhss-1265	2	32	a	a	DET
ajrhss-1265	2	33	differential	differential	ADJ
ajrhss-1265	2	34	equation	equation	NOUN
ajrhss-1265	2	35	.	.	PUNCT
ajrhss-1265	3	1	solving	solve	VERB
ajrhss-1265	3	2	such	such	ADJ
ajrhss-1265	3	3	equations	equation	NOUN
ajrhss-1265	3	4	often	often	ADV
ajrhss-1265	3	5	provides	provide	VERB
ajrhss-1265	3	6	information	information	NOUN
ajrhss-1265	3	7	about	about	ADP
ajrhss-1265	3	8	how	how	SCONJ
ajrhss-1265	3	9	quantities	quantity	NOUN
ajrhss-1265	3	10	change	change	VERB
ajrhss-1265	3	11	and	and	CCONJ
ajrhss-1265	3	12	frequently	frequently	ADV
ajrhss-1265	3	13	provides	provide	VERB
ajrhss-1265	3	14	insight	insight	NOUN
ajrhss-1265	3	15	into	into	ADP
ajrhss-1265	3	16	how	how	SCONJ
ajrhss-1265	3	17	and	and	CCONJ
ajrhss-1265	3	18	why	why	SCONJ
ajrhss-1265	3	19	the	the	DET
ajrhss-1265	3	20	changes	change	NOUN
ajrhss-1265	3	21	occur	occur	VERB
ajrhss-1265	3	22	.	.	PUNCT
ajrhss-1265	4	1	techniques	technique	NOUN
ajrhss-1265	4	2	for	for	ADP
ajrhss-1265	4	3	solving	solve	VERB
ajrhss-1265	4	4	differential	differential	ADJ
ajrhss-1265	4	5	equations	equation	NOUN
ajrhss-1265	4	6	can	can	AUX
ajrhss-1265	4	7	take	take	VERB
ajrhss-1265	4	8	many	many	ADJ
ajrhss-1265	4	9	different	different	ADJ
ajrhss-1265	4	10	forms	form	NOUN
ajrhss-1265	4	11	,	,	PUNCT
ajrhss-1265	4	12	including	include	VERB
ajrhss-1265	4	13	direct	direct	ADJ
ajrhss-1265	4	14	solution	solution	NOUN
ajrhss-1265	4	15	,	,	PUNCT
ajrhss-1265	4	16	use	use	NOUN
ajrhss-1265	4	17	of	of	ADP
ajrhss-1265	4	18	graphs	graph	NOUN
ajrhss-1265	4	19	,	,	PUNCT
ajrhss-1265	4	20	or	or	CCONJ
ajrhss-1265	4	21	computer	computer	NOUN
ajrhss-1265	4	22	calculations	calculation	NOUN
ajrhss-1265	4	23	.	.	PUNCT
ajrhss-1265	5	1	we	we	PRON
ajrhss-1265	5	2	introduce	introduce	VERB
ajrhss-1265	5	3	the	the	DET
ajrhss-1265	5	4	main	main	ADJ
ajrhss-1265	5	5	ideas	idea	NOUN
ajrhss-1265	5	6	in	in	ADP
ajrhss-1265	5	7	this	this	DET
ajrhss-1265	5	8	chapter	chapter	NOUN
ajrhss-1265	5	9	and	and	CCONJ
ajrhss-1265	5	10	describe	describe	VERB
ajrhss-1265	5	11	them	they	PRON
ajrhss-1265	5	12	in	in	ADP
ajrhss-1265	5	13	a	a	DET
ajrhss-1265	5	14	little	little	ADJ
ajrhss-1265	5	15	more	more	ADJ
ajrhss-1265	5	16	detail	detail	NOUN
ajrhss-1265	5	17	later	later	ADV
ajrhss-1265	5	18	in	in	ADP
ajrhss-1265	5	19	the	the	DET
ajrhss-1265	5	20	course	course	NOUN
ajrhss-1265	5	21	.	.	PUNCT
ajrhss-1265	6	1	in	in	ADP
ajrhss-1265	6	2	this	this	DET
ajrhss-1265	6	3	section	section	NOUN
ajrhss-1265	6	4	we	we	PRON
ajrhss-1265	6	5	study	study	VERB
ajrhss-1265	6	6	what	what	PRON
ajrhss-1265	6	7	differential	differential	ADJ
ajrhss-1265	6	8	equations	equation	NOUN
ajrhss-1265	6	9	are	be	AUX
ajrhss-1265	6	10	,	,	PUNCT
ajrhss-1265	6	11	how	how	SCONJ
ajrhss-1265	6	12	to	to	PART
ajrhss-1265	6	13	verify	verify	VERB
ajrhss-1265	6	14	their	their	PRON
ajrhss-1265	6	15	solutions	solution	NOUN
ajrhss-1265	6	16	,	,	PUNCT
ajrhss-1265	6	17	some	some	DET
ajrhss-1265	6	18	methods	method	NOUN
ajrhss-1265	6	19	that	that	PRON
ajrhss-1265	6	20	are	be	AUX
ajrhss-1265	6	21	used	use	VERB
ajrhss-1265	6	22	for	for	ADP
ajrhss-1265	6	23	solving	solve	VERB
ajrhss-1265	6	24	them	they	PRON
ajrhss-1265	6	25	,	,	PUNCT
ajrhss-1265	6	26	and	and	CCONJ
ajrhss-1265	6	27	some	some	DET
ajrhss-1265	6	28	examples	example	NOUN
ajrhss-1265	6	29	of	of	ADP
ajrhss-1265	6	30	common	common	ADJ
ajrhss-1265	6	31	and	and	CCONJ
ajrhss-1265	6	32	useful	useful	ADJ
ajrhss-1265	6	33	equations	equation	NOUN
ajrhss-1265	6	34	.	.	PUNCT
ajrhss-1265	7	1	differential	differential	ADJ
ajrhss-1265	7	2	equations	equation	NOUN
ajrhss-1265	7	3	,	,	PUNCT
ajrhss-1265	7	4	solutions	solution	NOUN
ajrhss-1265	7	5	,	,	PUNCT
ajrhss-1265	7	6	methods	method	NOUN
ajrhss-1265	7	7	,	,	PUNCT
ajrhss-1265	7	8	useful	useful	ADJ
ajrhss-1265	7	9	equations	equation	NOUN
ajrhss-1265	7	10	,	,	PUNCT
ajrhss-1265	7	11	specific	specific	ADJ
ajrhss-1265	7	12	information	information	NOUN
ajrhss-1265	7	13	.	.	PUNCT
ajrhss-1265	8	1	introduction	introduction	NOUN
ajrhss-1265	8	2	usually	usually	ADV
ajrhss-1265	8	3	a	a	DET
ajrhss-1265	8	4	given	give	VERB
ajrhss-1265	8	5	differential	differential	ADJ
ajrhss-1265	8	6	equation	equation	NOUN
ajrhss-1265	8	7	has	have	VERB
ajrhss-1265	8	8	an	an	DET
ajrhss-1265	8	9	infinite	infinite	ADJ
ajrhss-1265	8	10	number	number	NOUN
ajrhss-1265	8	11	of	of	ADP
ajrhss-1265	8	12	solutions	solution	NOUN
ajrhss-1265	8	13	,	,	PUNCT
ajrhss-1265	8	14	so	so	CCONJ
ajrhss-1265	8	15	it	it	PRON
ajrhss-1265	8	16	is	be	AUX
ajrhss-1265	8	17	natural	natural	ADJ
ajrhss-1265	8	18	to	to	PART
ajrhss-1265	8	19	ask	ask	VERB
ajrhss-1265	8	20	which	which	DET
ajrhss-1265	8	21	one	one	NOUN
ajrhss-1265	8	22	we	we	PRON
ajrhss-1265	8	23	want	want	VERB
ajrhss-1265	8	24	to	to	PART
ajrhss-1265	8	25	use	use	VERB
ajrhss-1265	8	26	.	.	PUNCT
ajrhss-1265	9	1	to	to	PART
ajrhss-1265	9	2	choose	choose	VERB
ajrhss-1265	9	3	one	one	NUM
ajrhss-1265	9	4	solution	solution	NOUN
ajrhss-1265	9	5	,	,	PUNCT
ajrhss-1265	9	6	more	more	ADJ
ajrhss-1265	9	7	information	information	NOUN
ajrhss-1265	9	8	is	be	AUX
ajrhss-1265	9	9	needed	need	VERB
ajrhss-1265	9	10	.	.	PUNCT
ajrhss-1265	10	1	some	some	DET
ajrhss-1265	10	2	specific	specific	ADJ
ajrhss-1265	10	3	information	information	NOUN
ajrhss-1265	10	4	that	that	PRON
ajrhss-1265	10	5	can	can	AUX
ajrhss-1265	10	6	be	be	AUX
ajrhss-1265	10	7	useful	useful	ADJ
ajrhss-1265	10	8	is	be	AUX
ajrhss-1265	10	9	an	an	DET
ajrhss-1265	10	10	initial	initial	ADJ
ajrhss-1265	10	11	value	value	NOUN
ajrhss-1265	10	12	,	,	PUNCT
ajrhss-1265	10	13	which	which	PRON
ajrhss-1265	10	14	is	be	AUX
ajrhss-1265	10	15	an	an	DET
ajrhss-1265	10	16	ordered	order	VERB
ajrhss-1265	10	17	pair	pair	NOUN
ajrhss-1265	10	18	that	that	PRON
ajrhss-1265	10	19	is	be	AUX
ajrhss-1265	10	20	used	use	VERB
ajrhss-1265	10	21	to	to	PART
ajrhss-1265	10	22	find	find	VERB
ajrhss-1265	10	23	a	a	DET
ajrhss-1265	10	24	particular	particular	ADJ
ajrhss-1265	10	25	solution	solution	NOUN
ajrhss-1265	10	26	.	.	PUNCT
ajrhss-1265	11	1	differential	differential	ADJ
ajrhss-1265	11	2	equation	equation	NOUN
ajrhss-1265	11	3	together	together	ADV
ajrhss-1265	11	4	with	with	ADP
ajrhss-1265	11	5	one	one	NUM
ajrhss-1265	11	6	or	or	CCONJ
ajrhss-1265	11	7	more	more	ADJ
ajrhss-1265	11	8	initial	initial	ADJ
ajrhss-1265	11	9	values	value	NOUN
ajrhss-1265	11	10	is	be	AUX
ajrhss-1265	11	11	called	call	VERB
ajrhss-1265	11	12	an	an	DET
ajrhss-1265	11	13	initial	initial	ADJ
ajrhss-1265	11	14	-	-	PUNCT
ajrhss-1265	11	15	value	value	NOUN
ajrhss-1265	11	16	problem	problem	NOUN
ajrhss-1265	11	17	.	.	PUNCT
ajrhss-1265	12	1	the	the	DET
ajrhss-1265	12	2	general	general	ADJ
ajrhss-1265	12	3	rule	rule	NOUN
ajrhss-1265	12	4	is	be	AUX
ajrhss-1265	12	5	that	that	SCONJ
ajrhss-1265	12	6	the	the	DET
ajrhss-1265	12	7	number	number	NOUN
ajrhss-1265	12	8	of	of	ADP
ajrhss-1265	12	9	initial	initial	ADJ
ajrhss-1265	12	10	values	value	NOUN
ajrhss-1265	12	11	needed	need	VERB
ajrhss-1265	12	12	for	for	ADP
ajrhss-1265	12	13	an	an	DET
ajrhss-1265	12	14	initial	initial	ADJ
ajrhss-1265	12	15	-	-	PUNCT
ajrhss-1265	12	16	value	value	NOUN
ajrhss-1265	12	17	problem	problem	NOUN
ajrhss-1265	12	18	is	be	AUX
ajrhss-1265	12	19	equal	equal	ADJ
ajrhss-1265	12	20	to	to	ADP
ajrhss-1265	12	21	the	the	DET
ajrhss-1265	12	22	order	order	NOUN
ajrhss-1265	12	23	of	of	ADP
ajrhss-1265	12	24	the	the	DET
ajrhss-1265	12	25	differential	differential	ADJ
ajrhss-1265	12	26	equation	equation	NOUN
ajrhss-1265	12	27	.	.	PUNCT
ajrhss-1265	13	1	for	for	ADP
ajrhss-1265	13	2	example	example	NOUN
ajrhss-1265	13	3	,	,	PUNCT
ajrhss-1265	13	4	if	if	SCONJ
ajrhss-1265	13	5	we	we	PRON
ajrhss-1265	13	6	have	have	VERB
ajrhss-1265	13	7	the	the	DET
ajrhss-1265	13	8	differential	differential	ADJ
ajrhss-1265	13	9	equation	equation	NOUN
ajrhss-1265	13	10	y'=2x	y'=2x	NOUN
ajrhss-1265	13	11	,	,	PUNCT
ajrhss-1265	13	12	then	then	ADV
ajrhss-1265	13	13	y	y	PROPN
ajrhss-1265	13	14	(	(	PUNCT
ajrhss-1265	13	15	3	3	X
ajrhss-1265	13	16	)	)	PUNCT
ajrhss-1265	13	17	=	=	SYM
ajrhss-1265	13	18	7	7	NUM
ajrhss-1265	13	19	is	be	AUX
ajrhss-1265	13	20	an	an	DET
ajrhss-1265	13	21	initial	initial	ADJ
ajrhss-1265	13	22	value	value	NOUN
ajrhss-1265	13	23	,	,	PUNCT
ajrhss-1265	13	24	and	and	CCONJ
ajrhss-1265	13	25	when	when	SCONJ
ajrhss-1265	13	26	taken	take	VERB
ajrhss-1265	13	27	together	together	ADV
ajrhss-1265	13	28	,	,	PUNCT
ajrhss-1265	13	29	these	these	DET
ajrhss-1265	13	30	equations	equation	NOUN
ajrhss-1265	13	31	form	form	VERB
ajrhss-1265	13	32	an	an	DET
ajrhss-1265	13	33	initial	initial	ADJ
ajrhss-1265	13	34	-	-	PUNCT
ajrhss-1265	13	35	value	value	NOUN
ajrhss-1265	13	36	problem	problem	NOUN
ajrhss-1265	13	37	?	?	PUNCT
ajrhss-1265	14	1	the	the	DET
ajrhss-1265	14	2	differential	differential	ADJ
ajrhss-1265	14	3	equation	equation	NOUN
ajrhss-1265	14	4	y′′−3y'+2y=4ex	y′′−3y'+2y=4ex	NOUN
ajrhss-1265	14	5	is	be	AUX
ajrhss-1265	14	6	second	second	ADJ
ajrhss-1265	14	7	order	order	NOUN
ajrhss-1265	14	8	,	,	PUNCT
ajrhss-1265	14	9	so	so	SCONJ
ajrhss-1265	14	10	we	we	PRON
ajrhss-1265	14	11	need	need	VERB
ajrhss-1265	14	12	two	two	NUM
ajrhss-1265	14	13	initial	initial	ADJ
ajrhss-1265	14	14	values	value	NOUN
ajrhss-1265	14	15	.	.	PUNCT
ajrhss-1265	15	1	with	with	ADP
ajrhss-1265	15	2	initial	initial	ADJ
ajrhss-1265	15	3	-	-	PUNCT
ajrhss-1265	15	4	value	value	NOUN
ajrhss-1265	15	5	problems	problem	NOUN
ajrhss-1265	15	6	of	of	ADP
ajrhss-1265	15	7	order	order	NOUN
ajrhss-1265	15	8	greater	great	ADJ
ajrhss-1265	15	9	than	than	ADP
ajrhss-1265	15	10	one	one	NUM
ajrhss-1265	15	11	,	,	PUNCT
ajrhss-1265	15	12	the	the	DET
ajrhss-1265	15	13	same	same	ADJ
ajrhss-1265	15	14	value	value	NOUN
ajrhss-1265	15	15	should	should	AUX
ajrhss-1265	15	16	be	be	AUX
ajrhss-1265	15	17	used	use	VERB
ajrhss-1265	15	18	for	for	ADP
ajrhss-1265	15	19	the	the	DET
ajrhss-1265	15	20	independent	independent	ADJ
ajrhss-1265	15	21	variable	variable	NOUN
ajrhss-1265	15	22	.	.	PUNCT
ajrhss-1265	16	1	an	an	DET
ajrhss-1265	16	2	example	example	NOUN
ajrhss-1265	16	3	of	of	ADP
ajrhss-1265	16	4	initial	initial	ADJ
ajrhss-1265	16	5	values	value	NOUN
ajrhss-1265	16	6	for	for	ADP
ajrhss-1265	16	7	this	this	DET
ajrhss-1265	16	8	second	second	ADJ
ajrhss-1265	16	9	-	-	PUNCT
ajrhss-1265	16	10	order	order	NOUN
ajrhss-1265	16	11	equation	equation	NOUN
ajrhss-1265	16	12	would	would	AUX
ajrhss-1265	16	13	be	be	AUX
ajrhss-1265	16	14	y	y	PROPN
ajrhss-1265	16	15	(	(	PUNCT
ajrhss-1265	16	16	0	0	NUM
ajrhss-1265	16	17	)	)	PUNCT
ajrhss-1265	16	18	=	=	SYM
ajrhss-1265	16	19	2	2	NUM
ajrhss-1265	16	20	and	and	CCONJ
ajrhss-1265	16	21	y'(0	y'(0	NOUN
ajrhss-1265	16	22	)	)	PUNCT
ajrhss-1265	17	1	=	=	SYM
ajrhss-1265	17	2	−1	−1	NOUN
ajrhss-1265	17	3	.	.	PUNCT
ajrhss-1265	18	1	these	these	DET
ajrhss-1265	18	2	two	two	NUM
ajrhss-1265	18	3	initial	initial	ADJ
ajrhss-1265	18	4	values	value	NOUN
ajrhss-1265	18	5	together	together	ADV
ajrhss-1265	18	6	with	with	ADP
ajrhss-1265	18	7	the	the	DET
ajrhss-1265	18	8	differential	differential	ADJ
ajrhss-1265	18	9	equation	equation	NOUN
ajrhss-1265	18	10	form	form	VERB
ajrhss-1265	18	11	an	an	DET
ajrhss-1265	18	12	initial	initial	ADJ
ajrhss-1265	18	13	-	-	PUNCT
ajrhss-1265	18	14	value	value	NOUN
ajrhss-1265	18	15	problem	problem	NOUN
ajrhss-1265	18	16	.	.	PUNCT
ajrhss-1265	19	1	these	these	DET
ajrhss-1265	19	2	problems	problem	NOUN
ajrhss-1265	19	3	are	be	AUX
ajrhss-1265	19	4	so	so	ADV
ajrhss-1265	19	5	named	name	VERB
ajrhss-1265	19	6	because	because	SCONJ
ajrhss-1265	19	7	often	often	ADV
ajrhss-1265	19	8	the	the	DET
ajrhss-1265	19	9	independent	independent	ADJ
ajrhss-1265	19	10	variable	variable	NOUN
ajrhss-1265	19	11	in	in	ADP
ajrhss-1265	19	12	the	the	DET
ajrhss-1265	19	13	unknown	unknown	ADJ
ajrhss-1265	19	14	function	function	NOUN
ajrhss-1265	19	15	is	be	AUX
ajrhss-1265	19	16	t	t	PROPN
ajrhss-1265	19	17	,	,	PUNCT
ajrhss-1265	19	18	which	which	PRON
ajrhss-1265	19	19	represents	represent	VERB
ajrhss-1265	19	20	time	time	NOUN
ajrhss-1265	19	21	.	.	PUNCT
ajrhss-1265	20	1	thus	thus	ADV
ajrhss-1265	20	2	,	,	PUNCT
ajrhss-1265	20	3	a	a	DET
ajrhss-1265	20	4	value	value	NOUN
ajrhss-1265	20	5	of	of	ADP
ajrhss-1265	20	6	t=0	t=0	PROPN
ajrhss-1265	20	7	represents	represent	VERB
ajrhss-1265	20	8	the	the	DET
ajrhss-1265	20	9	beginning	beginning	NOUN
ajrhss-1265	20	10	of	of	ADP
ajrhss-1265	20	11	the	the	DET
ajrhss-1265	20	12	problem	problem	NOUN
ajrhss-1265	20	13	.	.	PUNCT
ajrhss-1265	21	1	a	a	DET
ajrhss-1265	21	2	differential	differential	ADJ
ajrhss-1265	21	3	equation	equation	NOUN
ajrhss-1265	21	4	is	be	AUX
ajrhss-1265	21	5	an	an	DET
ajrhss-1265	21	6	equation	equation	NOUN
ajrhss-1265	21	7	involving	involve	VERB
ajrhss-1265	21	8	an	an	DET
ajrhss-1265	21	9	unknown	unknown	ADJ
ajrhss-1265	21	10	function	function	NOUN
ajrhss-1265	21	11	y	y	PROPN
ajrhss-1265	21	12	=	=	PROPN
ajrhss-1265	21	13	f(x	f(x	PROPN
ajrhss-1265	21	14	)	)	PUNCT
ajrhss-1265	21	15	and	and	CCONJ
ajrhss-1265	21	16	one	one	NUM
ajrhss-1265	21	17	or	or	CCONJ
ajrhss-1265	21	18	more	more	ADJ
ajrhss-1265	21	19	of	of	ADP
ajrhss-1265	21	20	its	its	PRON
ajrhss-1265	21	21	derivatives	derivative	NOUN
ajrhss-1265	21	22	.	.	PUNCT
ajrhss-1265	22	1	a	a	DET
ajrhss-1265	22	2	solution	solution	NOUN
ajrhss-1265	22	3	to	to	ADP
ajrhss-1265	22	4	a	a	DET
ajrhss-1265	22	5	differential	differential	ADJ
ajrhss-1265	22	6	equation	equation	NOUN
ajrhss-1265	22	7	is	be	AUX
ajrhss-1265	22	8	a	a	DET
ajrhss-1265	22	9	function	function	NOUN
ajrhss-1265	22	10	y	y	PROPN
ajrhss-1265	22	11	=	=	NOUN
ajrhss-1265	22	12	f(x	f(x	PROPN
ajrhss-1265	22	13	)	)	PUNCT
ajrhss-1265	22	14	that	that	PRON
ajrhss-1265	22	15	satisfies	satisfy	VERB
ajrhss-1265	22	16	the	the	DET
ajrhss-1265	22	17	differential	differential	ADJ
ajrhss-1265	22	18	equation	equation	NOUN
ajrhss-1265	22	19	when	when	SCONJ
ajrhss-1265	22	20	f	f	PROPN
ajrhss-1265	22	21	and	and	CCONJ
ajrhss-1265	22	22	its	its	PRON
ajrhss-1265	22	23	derivatives	derivative	NOUN
ajrhss-1265	22	24	are	be	AUX
ajrhss-1265	22	25	substituted	substitute	VERB
ajrhss-1265	22	26	into	into	ADP
ajrhss-1265	22	27	the	the	DET
ajrhss-1265	22	28	equation	equation	NOUN
ajrhss-1265	22	29	.	.	PUNCT
ajrhss-1265	23	1	we	we	PRON
ajrhss-1265	23	2	already	already	ADV
ajrhss-1265	23	3	noted	note	VERB
ajrhss-1265	23	4	that	that	SCONJ
ajrhss-1265	23	5	the	the	DET
ajrhss-1265	23	6	differential	differential	ADJ
ajrhss-1265	23	7	equation	equation	NOUN
ajrhss-1265	23	8	y'=2x	y'=2x	INTJ
ajrhss-1265	23	9	has	have	VERB
ajrhss-1265	23	10	at	at	ADV
ajrhss-1265	23	11	least	least	ADV
ajrhss-1265	23	12	two	two	NUM
ajrhss-1265	23	13	solutions	solution	NOUN
ajrhss-1265	23	14	:	:	PUNCT
ajrhss-1265	23	15	y	y	PROPN
ajrhss-1265	23	16	=	=	SYM
ajrhss-1265	23	17	x2	x2	PROPN
ajrhss-1265	23	18	and	and	CCONJ
ajrhss-1265	23	19	y	y	PROPN
ajrhss-1265	23	20	=	=	PROPN
ajrhss-1265	23	21	x2	x2	PROPN
ajrhss-1265	23	22	+	+	NOUN
ajrhss-1265	23	23	4	4	NUM
ajrhss-1265	23	24	.	.	PUNCT
ajrhss-1265	24	1	the	the	DET
ajrhss-1265	24	2	only	only	ADJ
ajrhss-1265	24	3	difference	difference	NOUN
ajrhss-1265	24	4	between	between	ADP
ajrhss-1265	24	5	these	these	DET
ajrhss-1265	24	6	two	two	NUM
ajrhss-1265	24	7	solutions	solution	NOUN
ajrhss-1265	24	8	is	be	AUX
ajrhss-1265	24	9	the	the	DET
ajrhss-1265	24	10	last	last	ADJ
ajrhss-1265	24	11	term	term	NOUN
ajrhss-1265	24	12	,	,	PUNCT
ajrhss-1265	24	13	which	which	PRON
ajrhss-1265	24	14	is	be	AUX
ajrhss-1265	24	15	a	a	DET
ajrhss-1265	24	16	constant	constant	ADJ
ajrhss-1265	24	17	.	.	PUNCT
ajrhss-1265	25	1	what	what	PRON
ajrhss-1265	25	2	if	if	SCONJ
ajrhss-1265	25	3	the	the	DET
ajrhss-1265	25	4	last	last	ADJ
ajrhss-1265	25	5	term	term	NOUN
ajrhss-1265	25	6	is	be	AUX
ajrhss-1265	25	7	a	a	DET
ajrhss-1265	25	8	different	different	ADJ
ajrhss-1265	25	9	constant	constant	ADJ
ajrhss-1265	25	10	?	?	PUNCT
ajrhss-1265	26	1	will	will	AUX
ajrhss-1265	26	2	this	this	DET
ajrhss-1265	26	3	expression	expression	NOUN
ajrhss-1265	26	4	still	still	ADV
ajrhss-1265	26	5	be	be	AUX
ajrhss-1265	26	6	a	a	DET
ajrhss-1265	26	7	solution	solution	NOUN
ajrhss-1265	26	8	to	to	ADP
ajrhss-1265	26	9	the	the	DET
ajrhss-1265	26	10	differential	differential	ADJ
ajrhss-1265	26	11	equation	equation	NOUN
ajrhss-1265	26	12	?	?	PUNCT
ajrhss-1265	27	1	in	in	ADP
ajrhss-1265	27	2	fact	fact	NOUN
ajrhss-1265	27	3	,	,	PUNCT
ajrhss-1265	27	4	any	any	DET
ajrhss-1265	27	5	function	function	NOUN
ajrhss-1265	27	6	of	of	ADP
ajrhss-1265	27	7	the	the	DET
ajrhss-1265	27	8	form	form	NOUN
ajrhss-1265	27	9	y	y	PROPN
ajrhss-1265	27	10	=	=	PROPN
ajrhss-1265	27	11	x2+c	x2+c	PROPN
ajrhss-1265	27	12	,	,	PUNCT
ajrhss-1265	27	13	where	where	SCONJ
ajrhss-1265	27	14	c	c	PROPN
ajrhss-1265	27	15	represents	represent	VERB
ajrhss-1265	27	16	any	any	DET
ajrhss-1265	27	17	constant	constant	ADJ
ajrhss-1265	27	18	,	,	PUNCT
ajrhss-1265	27	19	is	be	AUX
ajrhss-1265	27	20	a	a	DET
ajrhss-1265	27	21	american	american	ADJ
ajrhss-1265	27	22	journal	journal	NOUN
ajrhss-1265	27	23	of	of	ADP
ajrhss-1265	27	24	research	research	NOUN
ajrhss-1265	27	25	in	in	ADP
ajrhss-1265	27	26	humanities	humanity	NOUN
ajrhss-1265	27	27	and	and	CCONJ
ajrhss-1265	27	28	social	social	ADJ
ajrhss-1265	27	29	sciences	science	NOUN
ajrhss-1265	27	30	volume	volume	NOUN
ajrhss-1265	27	31	16	16	NUM
ajrhss-1265	27	32	sep	sep	PROPN
ajrhss-1265	27	33	.	.	PROPN
ajrhss-1265	27	34	,	,	PUNCT
ajrhss-1265	27	35	2023	2023	NUM
ajrhss-1265	27	36	p	p	NOUN
ajrhss-1265	27	37	a	a	DET
ajrhss-1265	27	38	g	g	NOUN
ajrhss-1265	27	39	e	e	PROPN
ajrhss-1265	27	40	|	|	ADV
ajrhss-1265	27	41	89	89	NUM
ajrhss-1265	27	42	www.americanjournal.org	www.americanjournal.org	NOUN
ajrhss-1265	27	43	solution	solution	NOUN
ajrhss-1265	27	44	as	as	ADV
ajrhss-1265	27	45	well	well	ADV
ajrhss-1265	27	46	.	.	PUNCT
ajrhss-1265	28	1	the	the	DET
ajrhss-1265	28	2	reason	reason	NOUN
ajrhss-1265	28	3	is	be	AUX
ajrhss-1265	28	4	that	that	SCONJ
ajrhss-1265	28	5	the	the	DET
ajrhss-1265	28	6	derivative	derivative	NOUN
ajrhss-1265	28	7	of	of	ADP
ajrhss-1265	28	8	x2+c	x2+c	PROPN
ajrhss-1265	28	9	is	be	AUX
ajrhss-1265	28	10	2x	2x	NUM
ajrhss-1265	28	11	,	,	PUNCT
ajrhss-1265	28	12	regardless	regardless	ADV
ajrhss-1265	28	13	of	of	ADP
ajrhss-1265	28	14	the	the	DET
ajrhss-1265	28	15	value	value	NOUN
ajrhss-1265	28	16	of	of	ADP
ajrhss-1265	28	17	c.	c.	NOUN
ajrhss-1265	28	18	it	it	PRON
ajrhss-1265	28	19	can	can	AUX
ajrhss-1265	28	20	be	be	AUX
ajrhss-1265	28	21	shown	show	VERB
ajrhss-1265	28	22	that	that	SCONJ
ajrhss-1265	28	23	any	any	DET
ajrhss-1265	28	24	solution	solution	NOUN
ajrhss-1265	28	25	of	of	ADP
ajrhss-1265	28	26	this	this	DET
ajrhss-1265	28	27	differential	differential	ADJ
ajrhss-1265	28	28	equation	equation	NOUN
ajrhss-1265	28	29	must	must	AUX
ajrhss-1265	28	30	be	be	AUX
ajrhss-1265	28	31	of	of	ADP
ajrhss-1265	28	32	the	the	DET
ajrhss-1265	28	33	form	form	NOUN
ajrhss-1265	28	34	y	y	PROPN
ajrhss-1265	28	35	=	=	PROPN
ajrhss-1265	28	36	x2+c	x2+c	PROPN
ajrhss-1265	28	37	this	this	PRON
ajrhss-1265	28	38	is	be	AUX
ajrhss-1265	28	39	an	an	DET
ajrhss-1265	28	40	example	example	NOUN
ajrhss-1265	28	41	of	of	ADP
ajrhss-1265	28	42	a	a	DET
ajrhss-1265	28	43	general	general	ADJ
ajrhss-1265	28	44	solution	solution	NOUN
ajrhss-1265	28	45	to	to	ADP
ajrhss-1265	28	46	a	a	DET
ajrhss-1265	28	47	differential	differential	ADJ
ajrhss-1265	28	48	equation	equation	NOUN
ajrhss-1265	28	49	.	.	PUNCT
ajrhss-1265	29	1	a	a	DET
ajrhss-1265	29	2	graph	graph	NOUN
ajrhss-1265	29	3	of	of	ADP
ajrhss-1265	29	4	some	some	PRON
ajrhss-1265	29	5	of	of	ADP
ajrhss-1265	29	6	these	these	DET
ajrhss-1265	29	7	solutions	solution	NOUN
ajrhss-1265	29	8	is	be	AUX
ajrhss-1265	29	9	given	give	VERB
ajrhss-1265	29	10	in	in	ADP
ajrhss-1265	29	11	figure	figure	NOUN
ajrhss-1265	29	12	1	1	NUM
ajrhss-1265	29	13	.	.	PUNCT
ajrhss-1265	30	1	(	(	PUNCT
ajrhss-1265	30	2	note	note	NOUN
ajrhss-1265	30	3	:	:	PUNCT
ajrhss-1265	30	4	in	in	ADP
ajrhss-1265	30	5	this	this	DET
ajrhss-1265	30	6	graph	graph	NOUN
ajrhss-1265	30	7	we	we	PRON
ajrhss-1265	30	8	used	use	VERB
ajrhss-1265	30	9	even	even	ADV
ajrhss-1265	30	10	integer	integer	NOUN
ajrhss-1265	30	11	values	value	NOUN
ajrhss-1265	30	12	for	for	ADP
ajrhss-1265	30	13	c	c	NOUN
ajrhss-1265	30	14	ranging	range	VERB
ajrhss-1265	30	15	between	between	ADP
ajrhss-1265	30	16	−4	−4	PROPN
ajrhss-1265	30	17	and	and	CCONJ
ajrhss-1265	30	18	4	4	NUM
ajrhss-1265	30	19	.	.	PUNCT
ajrhss-1265	31	1	in	in	ADP
ajrhss-1265	31	2	fact	fact	NOUN
ajrhss-1265	31	3	,	,	PUNCT
ajrhss-1265	31	4	there	there	PRON
ajrhss-1265	31	5	is	be	VERB
ajrhss-1265	31	6	no	no	DET
ajrhss-1265	31	7	restriction	restriction	NOUN
ajrhss-1265	31	8	on	on	ADP
ajrhss-1265	31	9	the	the	DET
ajrhss-1265	31	10	value	value	NOUN
ajrhss-1265	31	11	of	of	ADP
ajrhss-1265	31	12	c	c	NOUN
ajrhss-1265	31	13	;	;	PUNCT
ajrhss-1265	31	14	it	it	PRON
ajrhss-1265	31	15	can	can	AUX
ajrhss-1265	31	16	be	be	AUX
ajrhss-1265	31	17	an	an	DET
ajrhss-1265	31	18	integer	integer	NOUN
ajrhss-1265	31	19	or	or	CCONJ
ajrhss-1265	31	20	not	not	PART
ajrhss-1265	31	21	.	.	PUNCT
ajrhss-1265	31	22	)	)	PUNCT
ajrhss-1265	32	1	figure	figure	NOUN
ajrhss-1265	32	2	1	1	NUM
ajrhss-1265	32	3	.	.	PUNCT
ajrhss-1265	33	1	family	family	NOUN
ajrhss-1265	33	2	of	of	ADP
ajrhss-1265	33	3	solutions	solution	NOUN
ajrhss-1265	33	4	to	to	ADP
ajrhss-1265	33	5	the	the	DET
ajrhss-1265	33	6	differential	differential	ADJ
ajrhss-1265	33	7	equation	equation	NOUN
ajrhss-1265	33	8	y'=2x	y'=2x	INTJ
ajrhss-1265	33	9	in	in	ADP
ajrhss-1265	33	10	physics	physics	NOUN
ajrhss-1265	33	11	and	and	CCONJ
ajrhss-1265	33	12	engineering	engineering	NOUN
ajrhss-1265	33	13	applications	application	NOUN
ajrhss-1265	33	14	,	,	PUNCT
ajrhss-1265	33	15	we	we	PRON
ajrhss-1265	33	16	often	often	ADV
ajrhss-1265	33	17	consider	consider	VERB
ajrhss-1265	33	18	the	the	DET
ajrhss-1265	33	19	forces	force	NOUN
ajrhss-1265	33	20	acting	act	VERB
ajrhss-1265	33	21	upon	upon	SCONJ
ajrhss-1265	33	22	an	an	DET
ajrhss-1265	33	23	object	object	NOUN
ajrhss-1265	33	24	,	,	PUNCT
ajrhss-1265	33	25	and	and	CCONJ
ajrhss-1265	33	26	use	use	VERB
ajrhss-1265	33	27	this	this	DET
ajrhss-1265	33	28	information	information	NOUN
ajrhss-1265	33	29	to	to	PART
ajrhss-1265	33	30	understand	understand	VERB
ajrhss-1265	33	31	the	the	DET
ajrhss-1265	33	32	resulting	result	VERB
ajrhss-1265	33	33	motion	motion	NOUN
ajrhss-1265	33	34	that	that	PRON
ajrhss-1265	33	35	may	may	AUX
ajrhss-1265	33	36	occur	occur	VERB
ajrhss-1265	33	37	.	.	PUNCT
ajrhss-1265	34	1	for	for	ADP
ajrhss-1265	34	2	example	example	NOUN
ajrhss-1265	34	3	,	,	PUNCT
ajrhss-1265	34	4	if	if	SCONJ
ajrhss-1265	34	5	we	we	PRON
ajrhss-1265	34	6	start	start	VERB
ajrhss-1265	34	7	with	with	ADP
ajrhss-1265	34	8	an	an	DET
ajrhss-1265	34	9	object	object	NOUN
ajrhss-1265	34	10	at	at	ADP
ajrhss-1265	34	11	earth	earth	NOUN
ajrhss-1265	34	12	’s	’s	PART
ajrhss-1265	34	13	surface	surface	NOUN
ajrhss-1265	34	14	,	,	PUNCT
ajrhss-1265	34	15	the	the	DET
ajrhss-1265	34	16	primary	primary	ADJ
ajrhss-1265	34	17	force	force	NOUN
ajrhss-1265	34	18	acting	act	VERB
ajrhss-1265	34	19	upon	upon	SCONJ
ajrhss-1265	34	20	that	that	DET
ajrhss-1265	34	21	object	object	NOUN
ajrhss-1265	34	22	is	be	AUX
ajrhss-1265	34	23	gravity	gravity	NOUN
ajrhss-1265	34	24	.	.	PUNCT
ajrhss-1265	35	1	physicists	physicist	NOUN
ajrhss-1265	35	2	and	and	CCONJ
ajrhss-1265	35	3	engineers	engineer	NOUN
ajrhss-1265	35	4	can	can	AUX
ajrhss-1265	35	5	use	use	VERB
ajrhss-1265	35	6	this	this	DET
ajrhss-1265	35	7	information	information	NOUN
ajrhss-1265	35	8	,	,	PUNCT
ajrhss-1265	35	9	along	along	ADP
ajrhss-1265	35	10	with	with	ADP
ajrhss-1265	35	11	newton	newton	PROPN
ajrhss-1265	35	12	’s	’s	PART
ajrhss-1265	35	13	second	second	ADJ
ajrhss-1265	35	14	law	law	NOUN
ajrhss-1265	35	15	of	of	ADP
ajrhss-1265	35	16	motion	motion	NOUN
ajrhss-1265	35	17	(	(	PUNCT
ajrhss-1265	35	18	in	in	ADP
ajrhss-1265	35	19	equation	equation	NOUN
ajrhss-1265	35	20	form	form	NOUN
ajrhss-1265	35	21	f	f	PROPN
ajrhss-1265	35	22	=	=	X
ajrhss-1265	35	23	ma	ma	PROPN
ajrhss-1265	35	24	,	,	PUNCT
ajrhss-1265	35	25	where	where	SCONJ
ajrhss-1265	35	26	f	f	PROPN
ajrhss-1265	35	27	represents	represent	VERB
ajrhss-1265	35	28	force	force	NOUN
ajrhss-1265	35	29	,	,	PUNCT
ajrhss-1265	35	30	m	m	VERB
ajrhss-1265	35	31	represents	represent	VERB
ajrhss-1265	35	32	mass	mass	NOUN
ajrhss-1265	35	33	,	,	PUNCT
ajrhss-1265	35	34	and	and	CCONJ
ajrhss-1265	35	35	a	a	PRON
ajrhss-1265	35	36	represents	represent	VERB
ajrhss-1265	35	37	acceleration	acceleration	NOUN
ajrhss-1265	35	38	)	)	PUNCT
ajrhss-1265	35	39	,	,	PUNCT
ajrhss-1265	35	40	to	to	PART
ajrhss-1265	35	41	derive	derive	VERB
ajrhss-1265	35	42	an	an	DET
ajrhss-1265	35	43	equation	equation	NOUN
ajrhss-1265	35	44	that	that	PRON
ajrhss-1265	35	45	can	can	AUX
ajrhss-1265	35	46	be	be	AUX
ajrhss-1265	35	47	solved	solve	VERB
ajrhss-1265	35	48	.	.	PUNCT
ajrhss-1265	36	1	here	here	ADV
ajrhss-1265	36	2	are	be	AUX
ajrhss-1265	36	3	my	my	PRON
ajrhss-1265	36	4	notes	note	NOUN
ajrhss-1265	36	5	for	for	ADP
ajrhss-1265	36	6	my	my	PRON
ajrhss-1265	36	7	differential	differential	ADJ
ajrhss-1265	36	8	equations	equation	NOUN
ajrhss-1265	36	9	course	course	NOUN
ajrhss-1265	36	10	that	that	PRON
ajrhss-1265	36	11	i	i	PRON
ajrhss-1265	36	12	teach	teach	VERB
ajrhss-1265	36	13	here	here	ADV
ajrhss-1265	36	14	at	at	ADP
ajrhss-1265	36	15	lamar	lamar	PROPN
ajrhss-1265	36	16	university	university	PROPN
ajrhss-1265	36	17	.	.	PUNCT
ajrhss-1265	37	1	despite	despite	SCONJ
ajrhss-1265	37	2	the	the	DET
ajrhss-1265	37	3	fact	fact	NOUN
ajrhss-1265	37	4	that	that	SCONJ
ajrhss-1265	37	5	these	these	PRON
ajrhss-1265	37	6	are	be	AUX
ajrhss-1265	37	7	my	my	PRON
ajrhss-1265	37	8	“	"	PUNCT
ajrhss-1265	37	9	class	class	NOUN
ajrhss-1265	37	10	notes	note	NOUN
ajrhss-1265	37	11	”	"	PUNCT
ajrhss-1265	37	12	,	,	PUNCT
ajrhss-1265	37	13	they	they	PRON
ajrhss-1265	37	14	should	should	AUX
ajrhss-1265	37	15	be	be	AUX
ajrhss-1265	37	16	accessible	accessible	ADJ
ajrhss-1265	37	17	to	to	ADP
ajrhss-1265	37	18	anyone	anyone	PRON
ajrhss-1265	37	19	wanting	want	VERB
ajrhss-1265	37	20	to	to	PART
ajrhss-1265	37	21	learn	learn	VERB
ajrhss-1265	37	22	how	how	SCONJ
ajrhss-1265	37	23	to	to	PART
ajrhss-1265	37	24	solve	solve	VERB
ajrhss-1265	37	25	differential	differential	ADJ
ajrhss-1265	37	26	equations	equation	NOUN
ajrhss-1265	37	27	or	or	CCONJ
ajrhss-1265	37	28	needing	need	VERB
ajrhss-1265	37	29	a	a	DET
ajrhss-1265	37	30	refresher	refresher	NOUN
ajrhss-1265	37	31	on	on	ADP
ajrhss-1265	37	32	differential	differential	ADJ
ajrhss-1265	37	33	equations	equation	NOUN
ajrhss-1265	37	34	.	.	PUNCT
ajrhss-1265	38	1	i	i	PRON
ajrhss-1265	38	2	’ve	’ve	AUX
ajrhss-1265	38	3	tried	try	VERB
ajrhss-1265	38	4	to	to	PART
ajrhss-1265	38	5	make	make	VERB
ajrhss-1265	38	6	these	these	DET
ajrhss-1265	38	7	notes	note	NOUN
ajrhss-1265	38	8	as	as	SCONJ
ajrhss-1265	38	9	self	self	NOUN
ajrhss-1265	38	10	-	-	PUNCT
ajrhss-1265	38	11	contained	contain	VERB
ajrhss-1265	38	12	as	as	ADP
ajrhss-1265	38	13	possible	possible	ADJ
ajrhss-1265	38	14	and	and	CCONJ
ajrhss-1265	38	15	so	so	ADV
ajrhss-1265	38	16	all	all	DET
ajrhss-1265	38	17	the	the	DET
ajrhss-1265	38	18	information	information	NOUN
ajrhss-1265	38	19	needed	need	VERB
ajrhss-1265	38	20	to	to	PART
ajrhss-1265	38	21	read	read	VERB
ajrhss-1265	38	22	through	through	ADP
ajrhss-1265	38	23	them	they	PRON
ajrhss-1265	38	24	is	be	AUX
ajrhss-1265	38	25	either	either	CCONJ
ajrhss-1265	38	26	from	from	ADP
ajrhss-1265	38	27	a	a	DET
ajrhss-1265	38	28	calculus	calculus	NOUN
ajrhss-1265	38	29	or	or	CCONJ
ajrhss-1265	38	30	algebra	algebra	NOUN
ajrhss-1265	38	31	class	class	NOUN
ajrhss-1265	38	32	or	or	CCONJ
ajrhss-1265	38	33	contained	contain	VERB
ajrhss-1265	38	34	in	in	ADP
ajrhss-1265	38	35	other	other	ADJ
ajrhss-1265	38	36	sections	section	NOUN
ajrhss-1265	38	37	of	of	ADP
ajrhss-1265	38	38	the	the	DET
ajrhss-1265	38	39	notes	note	NOUN
ajrhss-1265	38	40	.	.	PUNCT
ajrhss-1265	39	1	an	an	DET
ajrhss-1265	39	2	ordinary	ordinary	ADJ
ajrhss-1265	39	3	differential	differential	ADJ
ajrhss-1265	39	4	equation	equation	NOUN
ajrhss-1265	39	5	(	(	PUNCT
ajrhss-1265	39	6	ode	ode	PROPN
ajrhss-1265	39	7	)	)	PUNCT
ajrhss-1265	39	8	is	be	AUX
ajrhss-1265	39	9	an	an	DET
ajrhss-1265	39	10	equation	equation	NOUN
ajrhss-1265	39	11	that	that	PRON
ajrhss-1265	39	12	involves	involve	VERB
ajrhss-1265	39	13	some	some	DET
ajrhss-1265	39	14	ordinary	ordinary	ADJ
ajrhss-1265	39	15	derivatives	derivative	NOUN
ajrhss-1265	39	16	(	(	PUNCT
ajrhss-1265	39	17	as	as	SCONJ
ajrhss-1265	39	18	opposed	oppose	VERB
ajrhss-1265	39	19	to	to	ADP
ajrhss-1265	39	20	partial	partial	ADJ
ajrhss-1265	39	21	derivatives	derivative	NOUN
ajrhss-1265	39	22	)	)	PUNCT
ajrhss-1265	39	23	of	of	ADP
ajrhss-1265	39	24	a	a	DET
ajrhss-1265	39	25	function	function	NOUN
ajrhss-1265	39	26	.	.	PUNCT
ajrhss-1265	40	1	often	often	ADV
ajrhss-1265	40	2	,	,	PUNCT
ajrhss-1265	40	3	our	our	PRON
ajrhss-1265	40	4	goal	goal	NOUN
ajrhss-1265	40	5	is	be	AUX
ajrhss-1265	40	6	to	to	PART
ajrhss-1265	40	7	solve	solve	VERB
ajrhss-1265	40	8	an	an	DET
ajrhss-1265	40	9	ode	ode	ADJ
ajrhss-1265	40	10	,	,	PUNCT
ajrhss-1265	40	11	i.e.	i.e.	X
ajrhss-1265	40	12	,	,	PUNCT
ajrhss-1265	40	13	determine	determine	VERB
ajrhss-1265	40	14	what	what	DET
ajrhss-1265	40	15	function	function	NOUN
ajrhss-1265	40	16	or	or	CCONJ
ajrhss-1265	40	17	functions	function	NOUN
ajrhss-1265	40	18	satisfy	satisfy	VERB
ajrhss-1265	40	19	the	the	DET
ajrhss-1265	40	20	equation	equation	NOUN
ajrhss-1265	40	21	.	.	PUNCT
ajrhss-1265	41	1	a	a	DET
ajrhss-1265	41	2	first	first	ADJ
ajrhss-1265	41	3	-	-	PUNCT
ajrhss-1265	41	4	order	order	NOUN
ajrhss-1265	41	5	differential	differential	ADJ
ajrhss-1265	41	6	equation	equation	NOUN
ajrhss-1265	41	7	is	be	AUX
ajrhss-1265	41	8	defined	define	VERB
ajrhss-1265	41	9	by	by	ADP
ajrhss-1265	41	10	an	an	DET
ajrhss-1265	41	11	equation	equation	NOUN
ajrhss-1265	41	12	:	:	PUNCT
ajrhss-1265	41	13	dy	dy	X
ajrhss-1265	41	14	/	/	SYM
ajrhss-1265	41	15	dx	dx	PROPN
ajrhss-1265	41	16	=	=	SYM
ajrhss-1265	41	17	f	f	PROPN
ajrhss-1265	41	18	(	(	PUNCT
ajrhss-1265	41	19	x	x	PROPN
ajrhss-1265	41	20	,	,	PUNCT
ajrhss-1265	41	21	y	y	NOUN
ajrhss-1265	41	22	)	)	PUNCT
ajrhss-1265	41	23	of	of	ADP
ajrhss-1265	41	24	two	two	NUM
ajrhss-1265	41	25	variables	variable	NOUN
ajrhss-1265	41	26	x	x	PUNCT
ajrhss-1265	41	27	and	and	CCONJ
ajrhss-1265	41	28	y	y	PROPN
ajrhss-1265	41	29	with	with	ADP
ajrhss-1265	41	30	its	its	PRON
ajrhss-1265	41	31	function	function	NOUN
ajrhss-1265	41	32	f(x	f(x	PROPN
ajrhss-1265	41	33	,	,	PUNCT
ajrhss-1265	41	34	y	y	NOUN
ajrhss-1265	41	35	)	)	PUNCT
ajrhss-1265	41	36	defined	define	VERB
ajrhss-1265	41	37	on	on	ADP
ajrhss-1265	41	38	a	a	DET
ajrhss-1265	41	39	region	region	NOUN
ajrhss-1265	41	40	in	in	ADP
ajrhss-1265	41	41	the	the	DET
ajrhss-1265	41	42	xy	xy	NOUN
ajrhss-1265	41	43	-	-	PUNCT
ajrhss-1265	41	44	plane	plane	NOUN
ajrhss-1265	41	45	.	.	PUNCT
ajrhss-1265	42	1	it	it	PRON
ajrhss-1265	42	2	has	have	VERB
ajrhss-1265	42	3	only	only	ADV
ajrhss-1265	42	4	the	the	DET
ajrhss-1265	42	5	first	first	ADJ
ajrhss-1265	42	6	derivative	derivative	ADJ
ajrhss-1265	42	7	dy	dy	NOUN
ajrhss-1265	42	8	/	/	SYM
ajrhss-1265	42	9	dx	dx	PROPN
ajrhss-1265	42	10	so	so	SCONJ
ajrhss-1265	42	11	that	that	SCONJ
ajrhss-1265	42	12	the	the	DET
ajrhss-1265	42	13	equation	equation	NOUN
ajrhss-1265	42	14	is	be	AUX
ajrhss-1265	42	15	of	of	ADP
ajrhss-1265	42	16	the	the	DET
ajrhss-1265	42	17	first	first	ADJ
ajrhss-1265	42	18	order	order	NOUN
ajrhss-1265	42	19	and	and	CCONJ
ajrhss-1265	42	20	no	no	DET
ajrhss-1265	42	21	higher	high	ADJ
ajrhss-1265	42	22	-	-	PUNCT
ajrhss-1265	42	23	order	order	NOUN
ajrhss-1265	42	24	derivatives	derivative	NOUN
ajrhss-1265	42	25	exist	exist	VERB
ajrhss-1265	42	26	.	.	PUNCT
ajrhss-1265	43	1	the	the	DET
ajrhss-1265	43	2	differential	differential	ADJ
ajrhss-1265	43	3	equation	equation	NOUN
ajrhss-1265	43	4	in	in	ADP
ajrhss-1265	43	5	first	first	ADJ
ajrhss-1265	43	6	-	-	PUNCT
ajrhss-1265	43	7	order	order	NOUN
ajrhss-1265	43	8	can	can	AUX
ajrhss-1265	43	9	also	also	ADV
ajrhss-1265	43	10	be	be	AUX
ajrhss-1265	43	11	written	write	VERB
ajrhss-1265	43	12	as	as	ADP
ajrhss-1265	43	13	;	;	PUNCT
ajrhss-1265	43	14	y	y	X
ajrhss-1265	43	15	’	'	PUNCT
ajrhss-1265	44	1	=	=	SYM
ajrhss-1265	44	2	f	f	X
ajrhss-1265	44	3	(	(	PUNCT
ajrhss-1265	44	4	x	x	PROPN
ajrhss-1265	44	5	,	,	PUNCT
ajrhss-1265	44	6	y	y	PROPN
ajrhss-1265	44	7	)	)	PUNCT
ajrhss-1265	44	8	or	or	CCONJ
ajrhss-1265	44	9	(	(	PUNCT
ajrhss-1265	44	10	d	d	X
ajrhss-1265	44	11	/	/	SYM
ajrhss-1265	44	12	dx	dx	PROPN
ajrhss-1265	44	13	)	)	PUNCT
ajrhss-1265	44	14	y	y	PROPN
ajrhss-1265	44	15	=	=	SYM
ajrhss-1265	44	16	f	f	PROPN
ajrhss-1265	44	17	(	(	PUNCT
ajrhss-1265	44	18	x	x	X
ajrhss-1265	44	19	,	,	PUNCT
ajrhss-1265	44	20	y	y	PROPN
ajrhss-1265	44	21	)	)	PUNCT
ajrhss-1265	44	22	the	the	DET
ajrhss-1265	44	23	differential	differential	ADJ
ajrhss-1265	44	24	equation	equation	NOUN
ajrhss-1265	44	25	is	be	AUX
ajrhss-1265	44	26	generally	generally	ADV
ajrhss-1265	44	27	used	use	VERB
ajrhss-1265	44	28	to	to	PART
ajrhss-1265	44	29	express	express	VERB
ajrhss-1265	44	30	a	a	DET
ajrhss-1265	44	31	relation	relation	NOUN
ajrhss-1265	44	32	between	between	ADP
ajrhss-1265	44	33	the	the	DET
ajrhss-1265	44	34	function	function	NOUN
ajrhss-1265	44	35	and	and	CCONJ
ajrhss-1265	44	36	its	its	PRON
ajrhss-1265	44	37	derivatives	derivative	NOUN
ajrhss-1265	44	38	.	.	PUNCT
ajrhss-1265	45	1	in	in	ADP
ajrhss-1265	45	2	physics	physics	NOUN
ajrhss-1265	45	3	and	and	CCONJ
ajrhss-1265	45	4	chemistry	chemistry	NOUN
ajrhss-1265	45	5	,	,	PUNCT
ajrhss-1265	45	6	it	it	PRON
ajrhss-1265	45	7	is	be	AUX
ajrhss-1265	45	8	used	use	VERB
ajrhss-1265	45	9	as	as	ADP
ajrhss-1265	45	10	a	a	DET
ajrhss-1265	45	11	technique	technique	NOUN
ajrhss-1265	45	12	for	for	ADP
ajrhss-1265	45	13	determining	determine	VERB
ajrhss-1265	45	14	the	the	DET
ajrhss-1265	45	15	functions	function	NOUN
ajrhss-1265	45	16	over	over	ADP
ajrhss-1265	45	17	its	its	PRON
ajrhss-1265	45	18	american	american	ADJ
ajrhss-1265	45	19	journal	journal	NOUN
ajrhss-1265	45	20	of	of	ADP
ajrhss-1265	45	21	research	research	NOUN
ajrhss-1265	45	22	in	in	ADP
ajrhss-1265	45	23	humanities	humanity	NOUN
ajrhss-1265	45	24	and	and	CCONJ
ajrhss-1265	45	25	social	social	ADJ
ajrhss-1265	45	26	sciences	science	NOUN
ajrhss-1265	45	27	volume	volume	NOUN
ajrhss-1265	45	28	16	16	NUM
ajrhss-1265	45	29	sep	sep	PROPN
ajrhss-1265	45	30	.	.	PROPN
ajrhss-1265	45	31	,	,	PUNCT
ajrhss-1265	45	32	2023	2023	NUM
ajrhss-1265	45	33	p	p	NOUN
ajrhss-1265	45	34	a	a	DET
ajrhss-1265	45	35	g	g	NOUN
ajrhss-1265	45	36	e	e	NOUN
ajrhss-1265	45	37	|	|	ADV
ajrhss-1265	45	38	90	90	NUM
ajrhss-1265	45	39	www.americanjournal.org	www.americanjournal.org	NOUN
ajrhss-1265	45	40	domain	domain	NOUN
ajrhss-1265	45	41	if	if	SCONJ
ajrhss-1265	45	42	we	we	PRON
ajrhss-1265	45	43	know	know	VERB
ajrhss-1265	45	44	the	the	DET
ajrhss-1265	45	45	functions	function	NOUN
ajrhss-1265	45	46	and	and	CCONJ
ajrhss-1265	45	47	some	some	PRON
ajrhss-1265	45	48	of	of	ADP
ajrhss-1265	45	49	the	the	DET
ajrhss-1265	45	50	derivatives	derivative	NOUN
ajrhss-1265	45	51	.	.	PUNCT
ajrhss-1265	46	1	if	if	SCONJ
ajrhss-1265	46	2	the	the	DET
ajrhss-1265	46	3	function	function	NOUN
ajrhss-1265	46	4	f	f	PROPN
ajrhss-1265	46	5	is	be	AUX
ajrhss-1265	46	6	a	a	DET
ajrhss-1265	46	7	linear	linear	ADJ
ajrhss-1265	46	8	expression	expression	NOUN
ajrhss-1265	46	9	in	in	ADP
ajrhss-1265	46	10	y	y	PROPN
ajrhss-1265	46	11	,	,	PUNCT
ajrhss-1265	46	12	then	then	ADV
ajrhss-1265	46	13	the	the	DET
ajrhss-1265	46	14	first	first	ADJ
ajrhss-1265	46	15	-	-	PUNCT
ajrhss-1265	46	16	order	order	NOUN
ajrhss-1265	46	17	differential	differential	ADJ
ajrhss-1265	46	18	equation	equation	NOUN
ajrhss-1265	46	19	y	y	NOUN
ajrhss-1265	46	20	’	'	PUNCT
ajrhss-1265	46	21	=	=	SYM
ajrhss-1265	47	1	f	f	X
ajrhss-1265	47	2	(	(	PUNCT
ajrhss-1265	47	3	x	x	NOUN
ajrhss-1265	47	4	,	,	PUNCT
ajrhss-1265	47	5	y	y	NOUN
ajrhss-1265	47	6	)	)	PUNCT
ajrhss-1265	47	7	is	be	AUX
ajrhss-1265	47	8	a	a	DET
ajrhss-1265	47	9	linear	linear	ADJ
ajrhss-1265	47	10	equation	equation	NOUN
ajrhss-1265	47	11	.	.	PUNCT
ajrhss-1265	48	1	that	that	PRON
ajrhss-1265	48	2	is	is	ADV
ajrhss-1265	48	3	,	,	PUNCT
ajrhss-1265	48	4	the	the	DET
ajrhss-1265	48	5	equation	equation	NOUN
ajrhss-1265	48	6	is	be	AUX
ajrhss-1265	48	7	linear	linear	ADJ
ajrhss-1265	48	8	and	and	CCONJ
ajrhss-1265	48	9	the	the	DET
ajrhss-1265	48	10	function	function	NOUN
ajrhss-1265	48	11	f	f	PROPN
ajrhss-1265	48	12	takes	take	VERB
ajrhss-1265	48	13	the	the	DET
ajrhss-1265	48	14	form	form	NOUN
ajrhss-1265	48	15	f(x	f(x	PROPN
ajrhss-1265	48	16	,	,	PUNCT
ajrhss-1265	48	17	y	y	PROPN
ajrhss-1265	48	18	)	)	PUNCT
ajrhss-1265	49	1	=	=	SYM
ajrhss-1265	49	2	p(x)y	p(x)y	PROPN
ajrhss-1265	49	3	+	+	NUM
ajrhss-1265	49	4	q(x	q(x	NOUN
ajrhss-1265	49	5	)	)	PUNCT
ajrhss-1265	49	6	where	where	SCONJ
ajrhss-1265	49	7	p	p	NOUN
ajrhss-1265	49	8	and	and	CCONJ
ajrhss-1265	49	9	q	q	NOUN
ajrhss-1265	49	10	are	be	AUX
ajrhss-1265	49	11	continuous	continuous	ADJ
ajrhss-1265	49	12	functions	function	NOUN
ajrhss-1265	49	13	on	on	ADP
ajrhss-1265	49	14	some	some	DET
ajrhss-1265	49	15	interval	interval	NOUN
ajrhss-1265	49	16	i.	i.	NOUN
ajrhss-1265	49	17	differential	differential	PROPN
ajrhss-1265	49	18	equations	equation	NOUN
ajrhss-1265	49	19	that	that	PRON
ajrhss-1265	49	20	are	be	AUX
ajrhss-1265	49	21	not	not	PART
ajrhss-1265	49	22	linear	linear	ADJ
ajrhss-1265	49	23	are	be	AUX
ajrhss-1265	49	24	called	call	VERB
ajrhss-1265	49	25	nonlinear	nonlinear	ADJ
ajrhss-1265	49	26	equations	equation	NOUN
ajrhss-1265	49	27	.	.	PUNCT
ajrhss-1265	50	1	this	this	DET
ajrhss-1265	50	2	method	method	NOUN
ajrhss-1265	50	3	is	be	AUX
ajrhss-1265	50	4	similar	similar	ADJ
ajrhss-1265	50	5	to	to	ADP
ajrhss-1265	50	6	the	the	DET
ajrhss-1265	50	7	integrating	integrate	VERB
ajrhss-1265	50	8	factor	factor	NOUN
ajrhss-1265	50	9	method	method	NOUN
ajrhss-1265	50	10	.	.	PUNCT
ajrhss-1265	51	1	finding	find	VERB
ajrhss-1265	51	2	the	the	DET
ajrhss-1265	51	3	general	general	ADJ
ajrhss-1265	51	4	solution	solution	NOUN
ajrhss-1265	51	5	of	of	ADP
ajrhss-1265	51	6	the	the	DET
ajrhss-1265	51	7	homogeneous	homogeneous	ADJ
ajrhss-1265	51	8	equation	equation	NOUN
ajrhss-1265	51	9	is	be	AUX
ajrhss-1265	51	10	the	the	DET
ajrhss-1265	51	11	first	first	ADJ
ajrhss-1265	51	12	necessary	necessary	ADJ
ajrhss-1265	51	13	step	step	NOUN
ajrhss-1265	51	14	.	.	PUNCT
ajrhss-1265	52	1	y	y	X
ajrhss-1265	52	2	’	'	PUNCT
ajrhss-1265	52	3	+	+	CCONJ
ajrhss-1265	52	4	a(x)y	a(x)y	PROPN
ajrhss-1265	52	5	=	=	SYM
ajrhss-1265	52	6	0	0	PROPN
ajrhss-1265	53	1	the	the	DET
ajrhss-1265	53	2	general	general	ADJ
ajrhss-1265	53	3	solution	solution	NOUN
ajrhss-1265	53	4	of	of	ADP
ajrhss-1265	53	5	the	the	DET
ajrhss-1265	53	6	homogeneous	homogeneous	ADJ
ajrhss-1265	53	7	equation	equation	NOUN
ajrhss-1265	53	8	always	always	ADV
ajrhss-1265	53	9	contains	contain	VERB
ajrhss-1265	53	10	a	a	DET
ajrhss-1265	53	11	constant	constant	NOUN
ajrhss-1265	53	12	of	of	ADP
ajrhss-1265	53	13	integration	integration	NOUN
ajrhss-1265	53	14	c.	c.	NOUN
ajrhss-1265	53	15	we	we	PRON
ajrhss-1265	53	16	can	can	AUX
ajrhss-1265	53	17	replace	replace	VERB
ajrhss-1265	53	18	the	the	DET
ajrhss-1265	53	19	constant	constant	ADJ
ajrhss-1265	53	20	c	c	NOUN
ajrhss-1265	53	21	with	with	ADP
ajrhss-1265	53	22	a	a	DET
ajrhss-1265	53	23	certain	certain	ADJ
ajrhss-1265	53	24	unknown	unknown	ADJ
ajrhss-1265	53	25	function	function	NOUN
ajrhss-1265	53	26	c(x	c(x	NOUN
ajrhss-1265	53	27	)	)	PUNCT
ajrhss-1265	53	28	.	.	PUNCT
ajrhss-1265	54	1	when	when	SCONJ
ajrhss-1265	54	2	substituting	substitute	VERB
ajrhss-1265	54	3	this	this	DET
ajrhss-1265	54	4	solution	solution	NOUN
ajrhss-1265	54	5	into	into	ADP
ajrhss-1265	54	6	the	the	DET
ajrhss-1265	54	7	non	non	ADJ
ajrhss-1265	54	8	-	-	ADJ
ajrhss-1265	54	9	homogeneous	homogeneous	ADJ
ajrhss-1265	54	10	differential	differential	NOUN
ajrhss-1265	54	11	equation	equation	NOUN
ajrhss-1265	54	12	,	,	PUNCT
ajrhss-1265	54	13	we	we	PRON
ajrhss-1265	54	14	can	can	AUX
ajrhss-1265	54	15	be	be	AUX
ajrhss-1265	54	16	able	able	ADJ
ajrhss-1265	54	17	to	to	PART
ajrhss-1265	54	18	determine	determine	VERB
ajrhss-1265	54	19	the	the	DET
ajrhss-1265	54	20	function	function	NOUN
ajrhss-1265	54	21	c(x	c(x	NOUN
ajrhss-1265	54	22	)	)	PUNCT
ajrhss-1265	54	23	.	.	PUNCT
ajrhss-1265	55	1	this	this	DET
ajrhss-1265	55	2	approach	approach	NOUN
ajrhss-1265	55	3	of	of	ADP
ajrhss-1265	55	4	the	the	DET
ajrhss-1265	55	5	algorithm	algorithm	NOUN
ajrhss-1265	55	6	is	be	AUX
ajrhss-1265	55	7	called	call	VERB
ajrhss-1265	55	8	the	the	DET
ajrhss-1265	55	9	method	method	NOUN
ajrhss-1265	55	10	of	of	ADP
ajrhss-1265	55	11	variation	variation	NOUN
ajrhss-1265	55	12	of	of	ADP
ajrhss-1265	55	13	a	a	DET
ajrhss-1265	55	14	constant	constant	ADJ
ajrhss-1265	55	15	.	.	PUNCT
ajrhss-1265	56	1	however	however	ADV
ajrhss-1265	56	2	,	,	PUNCT
ajrhss-1265	56	3	both	both	DET
ajrhss-1265	56	4	methods	method	NOUN
ajrhss-1265	56	5	lead	lead	VERB
ajrhss-1265	56	6	to	to	ADP
ajrhss-1265	56	7	the	the	DET
ajrhss-1265	56	8	same	same	ADJ
ajrhss-1265	56	9	solution	solution	NOUN
ajrhss-1265	56	10	.	.	PUNCT
ajrhss-1265	57	1	in	in	ADP
ajrhss-1265	57	2	calculus	calculus	NOUN
ajrhss-1265	57	3	,	,	PUNCT
ajrhss-1265	57	4	a	a	DET
ajrhss-1265	57	5	differential	differential	ADJ
ajrhss-1265	57	6	equation	equation	NOUN
ajrhss-1265	57	7	is	be	AUX
ajrhss-1265	57	8	an	an	DET
ajrhss-1265	57	9	equation	equation	NOUN
ajrhss-1265	57	10	that	that	PRON
ajrhss-1265	57	11	involves	involve	VERB
ajrhss-1265	57	12	the	the	DET
ajrhss-1265	57	13	derivative	derivative	ADJ
ajrhss-1265	57	14	(	(	PUNCT
ajrhss-1265	57	15	derivatives	derivative	NOUN
ajrhss-1265	57	16	)	)	PUNCT
ajrhss-1265	57	17	of	of	ADP
ajrhss-1265	57	18	the	the	DET
ajrhss-1265	57	19	dependent	dependent	ADJ
ajrhss-1265	57	20	variable	variable	NOUN
ajrhss-1265	57	21	with	with	ADP
ajrhss-1265	57	22	respect	respect	NOUN
ajrhss-1265	57	23	to	to	ADP
ajrhss-1265	57	24	the	the	DET
ajrhss-1265	57	25	independent	independent	ADJ
ajrhss-1265	57	26	variable	variable	NOUN
ajrhss-1265	57	27	(	(	PUNCT
ajrhss-1265	57	28	variables	variable	NOUN
ajrhss-1265	57	29	)	)	PUNCT
ajrhss-1265	57	30	.	.	PUNCT
ajrhss-1265	58	1	the	the	DET
ajrhss-1265	58	2	derivative	derivative	NOUN
ajrhss-1265	58	3	represents	represent	VERB
ajrhss-1265	58	4	nothing	nothing	PRON
ajrhss-1265	58	5	but	but	SCONJ
ajrhss-1265	58	6	a	a	DET
ajrhss-1265	58	7	rate	rate	NOUN
ajrhss-1265	58	8	of	of	ADP
ajrhss-1265	58	9	change	change	NOUN
ajrhss-1265	58	10	,	,	PUNCT
ajrhss-1265	58	11	and	and	CCONJ
ajrhss-1265	58	12	the	the	DET
ajrhss-1265	58	13	differential	differential	ADJ
ajrhss-1265	58	14	equation	equation	NOUN
ajrhss-1265	58	15	helps	help	VERB
ajrhss-1265	58	16	us	we	PRON
ajrhss-1265	58	17	present	present	VERB
ajrhss-1265	58	18	a	a	DET
ajrhss-1265	58	19	relationship	relationship	NOUN
ajrhss-1265	58	20	between	between	ADP
ajrhss-1265	58	21	the	the	DET
ajrhss-1265	58	22	changing	change	VERB
ajrhss-1265	58	23	quantity	quantity	NOUN
ajrhss-1265	58	24	with	with	ADP
ajrhss-1265	58	25	respect	respect	NOUN
ajrhss-1265	58	26	to	to	ADP
ajrhss-1265	58	27	the	the	DET
ajrhss-1265	58	28	change	change	NOUN
ajrhss-1265	58	29	in	in	ADP
ajrhss-1265	58	30	another	another	DET
ajrhss-1265	58	31	quantity	quantity	NOUN
ajrhss-1265	58	32	.	.	PUNCT
ajrhss-1265	59	1	y	y	X
ajrhss-1265	59	2	=	=	SYM
ajrhss-1265	59	3	f(x	f(x	PROPN
ajrhss-1265	59	4	)	)	PUNCT
ajrhss-1265	59	5	be	be	AUX
ajrhss-1265	59	6	a	a	DET
ajrhss-1265	59	7	function	function	NOUN
ajrhss-1265	59	8	where	where	SCONJ
ajrhss-1265	59	9	y	y	PROPN
ajrhss-1265	59	10	is	be	AUX
ajrhss-1265	59	11	a	a	DET
ajrhss-1265	59	12	dependent	dependent	ADJ
ajrhss-1265	59	13	variable	variable	NOUN
ajrhss-1265	59	14	,	,	PUNCT
ajrhss-1265	59	15	f	f	PROPN
ajrhss-1265	59	16	is	be	AUX
ajrhss-1265	59	17	an	an	DET
ajrhss-1265	59	18	unknown	unknown	ADJ
ajrhss-1265	59	19	function	function	NOUN
ajrhss-1265	59	20	,	,	PUNCT
ajrhss-1265	59	21	x	x	X
ajrhss-1265	59	22	is	be	AUX
ajrhss-1265	59	23	an	an	DET
ajrhss-1265	59	24	independent	independent	ADJ
ajrhss-1265	59	25	variable	variable	NOUN
ajrhss-1265	59	26	.	.	PUNCT
ajrhss-1265	60	1	here	here	ADV
ajrhss-1265	60	2	are	be	AUX
ajrhss-1265	60	3	a	a	DET
ajrhss-1265	60	4	few	few	ADJ
ajrhss-1265	60	5	differential	differential	ADJ
ajrhss-1265	60	6	equations	equation	NOUN
ajrhss-1265	60	7	.	.	PUNCT
ajrhss-1265	61	1	(	(	PUNCT
ajrhss-1265	61	2	dy	dy	X
ajrhss-1265	61	3	/	/	SYM
ajrhss-1265	61	4	dx	dx	PROPN
ajrhss-1265	61	5	)	)	PUNCT
ajrhss-1265	62	1	=	=	PUNCT
ajrhss-1265	62	2	sin	sin	NOUN
ajrhss-1265	62	3	x	x	SYM
ajrhss-1265	62	4	(	(	PUNCT
ajrhss-1265	62	5	d2y	d2y	NOUN
ajrhss-1265	62	6	/	/	SYM
ajrhss-1265	62	7	dx2	dx2	NOUN
ajrhss-1265	62	8	)	)	PUNCT
ajrhss-1265	63	1	+	+	CCONJ
ajrhss-1265	64	1	k2y	k2y	PROPN
ajrhss-1265	64	2	=	=	SYM
ajrhss-1265	64	3	0	0	NUM
ajrhss-1265	64	4	(	(	PUNCT
ajrhss-1265	64	5	d2y	d2y	NOUN
ajrhss-1265	64	6	/	/	SYM
ajrhss-1265	64	7	dt2	dt2	NOUN
ajrhss-1265	64	8	)	)	PUNCT
ajrhss-1265	64	9	+	+	CCONJ
ajrhss-1265	64	10	(	(	PUNCT
ajrhss-1265	64	11	d2x	d2x	PROPN
ajrhss-1265	64	12	/	/	SYM
ajrhss-1265	64	13	dt2	dt2	PROPN
ajrhss-1265	64	14	)	)	PUNCT
ajrhss-1265	64	15	=	=	SYM
ajrhss-1265	64	16	x	x	X
ajrhss-1265	64	17	(	(	PUNCT
ajrhss-1265	64	18	d3y	d3y	PROPN
ajrhss-1265	64	19	/	/	SYM
ajrhss-1265	64	20	dx3	dx3	NOUN
ajrhss-1265	64	21	)	)	PUNCT
ajrhss-1265	64	22	+	+	CCONJ
ajrhss-1265	64	23	x(dy	x(dy	PROPN
ajrhss-1265	64	24	/	/	SYM
ajrhss-1265	64	25	dx	dx	PROPN
ajrhss-1265	64	26	)	)	PUNCT
ajrhss-1265	64	27	4xy	4xy	NOUN
ajrhss-1265	65	1	=	=	SYM
ajrhss-1265	65	2	0	0	NUM
ajrhss-1265	65	3	(	(	PUNCT
ajrhss-1265	65	4	rdr	rdr	PROPN
ajrhss-1265	65	5	/	/	SYM
ajrhss-1265	65	6	dθ	dθ	PROPN
ajrhss-1265	65	7	)	)	PUNCT
ajrhss-1265	66	1	+	+	CCONJ
ajrhss-1265	66	2	cosθ	cosθ	X
ajrhss-1265	66	3	=	=	SYM
ajrhss-1265	66	4	5	5	NUM
ajrhss-1265	66	5	american	american	ADJ
ajrhss-1265	66	6	journal	journal	NOUN
ajrhss-1265	66	7	of	of	ADP
ajrhss-1265	66	8	research	research	NOUN
ajrhss-1265	66	9	in	in	ADP
ajrhss-1265	66	10	humanities	humanity	NOUN
ajrhss-1265	66	11	and	and	CCONJ
ajrhss-1265	66	12	social	social	ADJ
ajrhss-1265	66	13	sciences	science	NOUN
ajrhss-1265	66	14	volume	volume	NOUN
ajrhss-1265	66	15	16	16	NUM
ajrhss-1265	66	16	sep	sep	PROPN
ajrhss-1265	66	17	.	.	PROPN
ajrhss-1265	66	18	,	,	PUNCT
ajrhss-1265	66	19	2023	2023	NUM
ajrhss-1265	66	20	p	p	NOUN
ajrhss-1265	66	21	a	a	DET
ajrhss-1265	66	22	g	g	NOUN
ajrhss-1265	66	23	e	e	NOUN
ajrhss-1265	66	24	|	|	ADV
ajrhss-1265	66	25	91	91	NUM
ajrhss-1265	66	26	www.americanjournal.org	www.americanjournal.org	NOUN
ajrhss-1265	66	27	figure	figure	NOUN
ajrhss-1265	66	28	2	2	NUM
ajrhss-1265	66	29	.	.	PUNCT
ajrhss-1265	66	30	differential	differential	ADJ
ajrhss-1265	66	31	equation	equation	NOUN
ajrhss-1265	66	32	.	.	PUNCT
ajrhss-1265	67	1	if	if	SCONJ
ajrhss-1265	67	2	a	a	DET
ajrhss-1265	67	3	differential	differential	ADJ
ajrhss-1265	67	4	equation	equation	NOUN
ajrhss-1265	67	5	is	be	AUX
ajrhss-1265	67	6	expressible	expressible	ADJ
ajrhss-1265	67	7	in	in	ADP
ajrhss-1265	67	8	a	a	DET
ajrhss-1265	67	9	polynomial	polynomial	ADJ
ajrhss-1265	67	10	form	form	NOUN
ajrhss-1265	67	11	,	,	PUNCT
ajrhss-1265	67	12	then	then	ADV
ajrhss-1265	67	13	the	the	DET
ajrhss-1265	67	14	integral	integral	ADJ
ajrhss-1265	67	15	power	power	NOUN
ajrhss-1265	67	16	of	of	ADP
ajrhss-1265	67	17	the	the	DET
ajrhss-1265	67	18	highest	high	ADJ
ajrhss-1265	67	19	order	order	NOUN
ajrhss-1265	67	20	derivative	derivative	NOUN
ajrhss-1265	67	21	that	that	PRON
ajrhss-1265	67	22	appears	appear	VERB
ajrhss-1265	67	23	is	be	AUX
ajrhss-1265	67	24	called	call	VERB
ajrhss-1265	67	25	the	the	DET
ajrhss-1265	67	26	degree	degree	NOUN
ajrhss-1265	67	27	of	of	ADP
ajrhss-1265	67	28	the	the	DET
ajrhss-1265	67	29	differential	differential	ADJ
ajrhss-1265	67	30	equation	equation	NOUN
ajrhss-1265	67	31	.	.	PUNCT
ajrhss-1265	68	1	the	the	DET
ajrhss-1265	68	2	degree	degree	NOUN
ajrhss-1265	68	3	of	of	ADP
ajrhss-1265	68	4	the	the	DET
ajrhss-1265	68	5	differential	differential	ADJ
ajrhss-1265	68	6	equation	equation	NOUN
ajrhss-1265	68	7	is	be	AUX
ajrhss-1265	68	8	the	the	DET
ajrhss-1265	68	9	power	power	NOUN
ajrhss-1265	68	10	of	of	ADP
ajrhss-1265	68	11	the	the	DET
ajrhss-1265	68	12	highest	high	ADJ
ajrhss-1265	68	13	ordered	order	VERB
ajrhss-1265	68	14	derivative	derivative	ADJ
ajrhss-1265	68	15	present	present	NOUN
ajrhss-1265	68	16	in	in	ADP
ajrhss-1265	68	17	the	the	DET
ajrhss-1265	68	18	equation	equation	NOUN
ajrhss-1265	68	19	.	.	PUNCT
ajrhss-1265	69	1	to	to	PART
ajrhss-1265	69	2	find	find	VERB
ajrhss-1265	69	3	the	the	DET
ajrhss-1265	69	4	degree	degree	NOUN
ajrhss-1265	69	5	of	of	ADP
ajrhss-1265	69	6	the	the	DET
ajrhss-1265	69	7	differential	differential	ADJ
ajrhss-1265	69	8	equation	equation	NOUN
ajrhss-1265	69	9	,	,	PUNCT
ajrhss-1265	69	10	we	we	PRON
ajrhss-1265	69	11	need	need	VERB
ajrhss-1265	69	12	to	to	PART
ajrhss-1265	69	13	have	have	VERB
ajrhss-1265	69	14	a	a	DET
ajrhss-1265	69	15	positive	positive	ADJ
ajrhss-1265	69	16	integer	integer	NOUN
ajrhss-1265	69	17	as	as	ADP
ajrhss-1265	69	18	the	the	DET
ajrhss-1265	69	19	index	index	NOUN
ajrhss-1265	69	20	of	of	ADP
ajrhss-1265	69	21	each	each	DET
ajrhss-1265	69	22	derivative	derivative	NOUN
ajrhss-1265	69	23	.	.	PUNCT
ajrhss-1265	70	1	a	a	DET
ajrhss-1265	70	2	differential	differential	ADJ
ajrhss-1265	70	3	equation	equation	NOUN
ajrhss-1265	70	4	in	in	ADP
ajrhss-1265	70	5	which	which	PRON
ajrhss-1265	70	6	the	the	DET
ajrhss-1265	70	7	degree	degree	NOUN
ajrhss-1265	70	8	of	of	ADP
ajrhss-1265	70	9	all	all	DET
ajrhss-1265	70	10	the	the	DET
ajrhss-1265	70	11	terms	term	NOUN
ajrhss-1265	70	12	is	be	AUX
ajrhss-1265	70	13	the	the	DET
ajrhss-1265	70	14	same	same	ADJ
ajrhss-1265	70	15	is	be	AUX
ajrhss-1265	70	16	known	know	VERB
ajrhss-1265	70	17	as	as	ADP
ajrhss-1265	70	18	a	a	DET
ajrhss-1265	70	19	homogenous	homogenous	ADJ
ajrhss-1265	70	20	differential	differential	NOUN
ajrhss-1265	70	21	equation	equation	NOUN
ajrhss-1265	70	22	.	.	PUNCT
ajrhss-1265	71	1	in	in	ADP
ajrhss-1265	71	2	general	general	ADJ
ajrhss-1265	71	3	they	they	PRON
ajrhss-1265	71	4	can	can	AUX
ajrhss-1265	71	5	be	be	AUX
ajrhss-1265	71	6	represented	represent	VERB
ajrhss-1265	71	7	as	as	ADP
ajrhss-1265	71	8	p(x	p(x	NOUN
ajrhss-1265	71	9	,	,	PUNCT
ajrhss-1265	71	10	y)dx	y)dx	PROPN
ajrhss-1265	71	11	+	+	NUM
ajrhss-1265	71	12	q(x	q(x	NOUN
ajrhss-1265	71	13	,	,	PUNCT
ajrhss-1265	71	14	y)dy	y)dy	PROPN
ajrhss-1265	71	15	=	=	SYM
ajrhss-1265	71	16	0	0	NUM
ajrhss-1265	71	17	,	,	PUNCT
ajrhss-1265	71	18	where	where	SCONJ
ajrhss-1265	71	19	p(x	p(x	PROPN
ajrhss-1265	71	20	,	,	PUNCT
ajrhss-1265	71	21	y	y	NOUN
ajrhss-1265	71	22	)	)	PUNCT
ajrhss-1265	71	23	and	and	CCONJ
ajrhss-1265	71	24	q(x	q(x	PROPN
ajrhss-1265	71	25	,	,	PUNCT
ajrhss-1265	71	26	y	y	NOUN
ajrhss-1265	71	27	)	)	PUNCT
ajrhss-1265	71	28	are	be	AUX
ajrhss-1265	71	29	homogeneous	homogeneous	ADJ
ajrhss-1265	71	30	functions	function	NOUN
ajrhss-1265	71	31	of	of	ADP
ajrhss-1265	71	32	the	the	DET
ajrhss-1265	71	33	same	same	ADJ
ajrhss-1265	71	34	degree	degree	NOUN
ajrhss-1265	71	35	.	.	PUNCT
ajrhss-1265	72	1	the	the	DET
ajrhss-1265	72	2	differential	differential	ADJ
ajrhss-1265	72	3	equation	equation	NOUN
ajrhss-1265	72	4	has	have	VERB
ajrhss-1265	72	5	infinitely	infinitely	ADV
ajrhss-1265	72	6	many	many	ADJ
ajrhss-1265	72	7	solutions	solution	NOUN
ajrhss-1265	72	8	.	.	PUNCT
ajrhss-1265	73	1	solving	solve	VERB
ajrhss-1265	73	2	a	a	DET
ajrhss-1265	73	3	differential	differential	ADJ
ajrhss-1265	73	4	equation	equation	NOUN
ajrhss-1265	73	5	is	be	AUX
ajrhss-1265	73	6	referred	refer	VERB
ajrhss-1265	73	7	to	to	ADP
ajrhss-1265	73	8	as	as	ADP
ajrhss-1265	73	9	integrating	integrate	VERB
ajrhss-1265	73	10	a	a	DET
ajrhss-1265	73	11	differential	differential	ADJ
ajrhss-1265	73	12	equation	equation	NOUN
ajrhss-1265	73	13	since	since	SCONJ
ajrhss-1265	73	14	the	the	DET
ajrhss-1265	73	15	process	process	NOUN
ajrhss-1265	73	16	of	of	ADP
ajrhss-1265	73	17	finding	find	VERB
ajrhss-1265	73	18	the	the	DET
ajrhss-1265	73	19	solution	solution	NOUN
ajrhss-1265	73	20	to	to	ADP
ajrhss-1265	73	21	a	a	DET
ajrhss-1265	73	22	differential	differential	ADJ
ajrhss-1265	73	23	equation	equation	NOUN
ajrhss-1265	73	24	involves	involve	VERB
ajrhss-1265	73	25	integration	integration	NOUN
ajrhss-1265	73	26	.	.	PUNCT
ajrhss-1265	74	1	a	a	DET
ajrhss-1265	74	2	solution	solution	NOUN
ajrhss-1265	74	3	of	of	ADP
ajrhss-1265	74	4	a	a	DET
ajrhss-1265	74	5	differential	differential	ADJ
ajrhss-1265	74	6	equation	equation	NOUN
ajrhss-1265	74	7	is	be	AUX
ajrhss-1265	74	8	an	an	DET
ajrhss-1265	74	9	expression	expression	NOUN
ajrhss-1265	74	10	for	for	ADP
ajrhss-1265	74	11	the	the	DET
ajrhss-1265	74	12	dependent	dependent	ADJ
ajrhss-1265	74	13	variable	variable	NOUN
ajrhss-1265	74	14	in	in	ADP
ajrhss-1265	74	15	terms	term	NOUN
ajrhss-1265	74	16	of	of	ADP
ajrhss-1265	74	17	the	the	DET
ajrhss-1265	74	18	independent	independent	ADJ
ajrhss-1265	74	19	variable	variable	NOUN
ajrhss-1265	74	20	which	which	PRON
ajrhss-1265	74	21	satisfies	satisfy	VERB
ajrhss-1265	74	22	the	the	DET
ajrhss-1265	74	23	differential	differential	ADJ
ajrhss-1265	74	24	equation	equation	NOUN
ajrhss-1265	74	25	.	.	PUNCT
ajrhss-1265	75	1	the	the	DET
ajrhss-1265	75	2	solution	solution	NOUN
ajrhss-1265	75	3	which	which	PRON
ajrhss-1265	75	4	contains	contain	VERB
ajrhss-1265	75	5	as	as	SCONJ
ajrhss-1265	75	6	many	many	ADJ
ajrhss-1265	75	7	arbitrary	arbitrary	ADJ
ajrhss-1265	75	8	constants	constant	NOUN
ajrhss-1265	75	9	is	be	AUX
ajrhss-1265	75	10	called	call	VERB
ajrhss-1265	75	11	the	the	DET
ajrhss-1265	75	12	general	general	ADJ
ajrhss-1265	75	13	solution	solution	NOUN
ajrhss-1265	75	14	.	.	PUNCT
ajrhss-1265	76	1	if	if	SCONJ
ajrhss-1265	76	2	we	we	PRON
ajrhss-1265	76	3	give	give	VERB
ajrhss-1265	76	4	particular	particular	ADJ
ajrhss-1265	76	5	values	value	NOUN
ajrhss-1265	76	6	to	to	ADP
ajrhss-1265	76	7	the	the	DET
ajrhss-1265	76	8	arbitrary	arbitrary	ADJ
ajrhss-1265	76	9	constants	constant	NOUN
ajrhss-1265	76	10	in	in	ADP
ajrhss-1265	76	11	the	the	DET
ajrhss-1265	76	12	general	general	ADJ
ajrhss-1265	76	13	solution	solution	NOUN
ajrhss-1265	76	14	of	of	ADP
ajrhss-1265	76	15	the	the	DET
ajrhss-1265	76	16	differential	differential	ADJ
ajrhss-1265	76	17	equation	equation	NOUN
ajrhss-1265	76	18	,	,	PUNCT
ajrhss-1265	76	19	the	the	DET
ajrhss-1265	76	20	resulting	result	VERB
ajrhss-1265	76	21	solution	solution	NOUN
ajrhss-1265	76	22	is	be	AUX
ajrhss-1265	76	23	called	call	VERB
ajrhss-1265	76	24	a	a	DET
ajrhss-1265	76	25	particular	particular	ADJ
ajrhss-1265	76	26	solution	solution	NOUN
ajrhss-1265	76	27	.	.	PUNCT
ajrhss-1265	77	1	the	the	DET
ajrhss-1265	77	2	result	result	NOUN
ajrhss-1265	77	3	of	of	ADP
ajrhss-1265	77	4	eliminating	eliminate	VERB
ajrhss-1265	77	5	one	one	NUM
ajrhss-1265	77	6	arbitrary	arbitrary	ADJ
ajrhss-1265	77	7	constant	constant	ADJ
ajrhss-1265	77	8	yields	yield	NOUN
ajrhss-1265	77	9	a	a	DET
ajrhss-1265	77	10	first	first	ADJ
ajrhss-1265	77	11	-	-	PUNCT
ajrhss-1265	77	12	order	order	NOUN
ajrhss-1265	77	13	differential	differential	ADJ
ajrhss-1265	77	14	equation	equation	NOUN
ajrhss-1265	77	15	and	and	CCONJ
ajrhss-1265	77	16	that	that	PRON
ajrhss-1265	77	17	of	of	ADP
ajrhss-1265	77	18	eliminating	eliminate	VERB
ajrhss-1265	77	19	two	two	NUM
ajrhss-1265	77	20	arbitrary	arbitrary	ADJ
ajrhss-1265	77	21	constants	constant	NOUN
ajrhss-1265	77	22	leads	lead	VERB
ajrhss-1265	77	23	to	to	ADP
ajrhss-1265	77	24	a	a	DET
ajrhss-1265	77	25	second	second	ADJ
ajrhss-1265	77	26	-	-	PUNCT
ajrhss-1265	77	27	order	order	NOUN
ajrhss-1265	77	28	differential	differential	ADJ
ajrhss-1265	77	29	equation	equation	NOUN
ajrhss-1265	77	30	and	and	CCONJ
ajrhss-1265	77	31	so	so	ADV
ajrhss-1265	77	32	on	on	ADV
ajrhss-1265	77	33	.	.	PUNCT
ajrhss-1265	78	1	let	let	VERB
ajrhss-1265	78	2	us	we	PRON
ajrhss-1265	78	3	understand	understand	VERB
ajrhss-1265	78	4	solving	solve	VERB
ajrhss-1265	78	5	the	the	DET
ajrhss-1265	78	6	differential	differential	ADJ
ajrhss-1265	78	7	equation	equation	NOUN
ajrhss-1265	78	8	by	by	ADP
ajrhss-1265	78	9	an	an	DET
ajrhss-1265	78	10	example	example	NOUN
ajrhss-1265	78	11	.	.	PUNCT
ajrhss-1265	79	1	ordinary	ordinary	ADJ
ajrhss-1265	79	2	differential	differential	ADJ
ajrhss-1265	79	3	equations	equation	NOUN
ajrhss-1265	79	4	applications	application	NOUN
ajrhss-1265	79	5	in	in	ADP
ajrhss-1265	79	6	real	real	ADJ
ajrhss-1265	79	7	life	life	NOUN
ajrhss-1265	79	8	are	be	AUX
ajrhss-1265	79	9	used	use	VERB
ajrhss-1265	79	10	to	to	PART
ajrhss-1265	79	11	calculate	calculate	VERB
ajrhss-1265	79	12	the	the	DET
ajrhss-1265	79	13	movement	movement	NOUN
ajrhss-1265	79	14	or	or	CCONJ
ajrhss-1265	79	15	flow	flow	NOUN
ajrhss-1265	79	16	of	of	ADP
ajrhss-1265	79	17	electricity	electricity	NOUN
ajrhss-1265	79	18	,	,	PUNCT
ajrhss-1265	79	19	motion	motion	NOUN
ajrhss-1265	79	20	of	of	ADP
ajrhss-1265	79	21	an	an	DET
ajrhss-1265	79	22	object	object	NOUN
ajrhss-1265	79	23	to	to	ADP
ajrhss-1265	79	24	and	and	CCONJ
ajrhss-1265	79	25	fro	fro	VERB
ajrhss-1265	79	26	like	like	ADP
ajrhss-1265	79	27	a	a	DET
ajrhss-1265	79	28	pendulum	pendulum	NOUN
ajrhss-1265	79	29	,	,	PUNCT
ajrhss-1265	79	30	to	to	PART
ajrhss-1265	79	31	explain	explain	VERB
ajrhss-1265	79	32	thermodynamics	thermodynamic	NOUN
ajrhss-1265	79	33	concepts	concept	NOUN
ajrhss-1265	79	34	.	.	PUNCT
ajrhss-1265	80	1	also	also	ADV
ajrhss-1265	80	2	,	,	PUNCT
ajrhss-1265	80	3	in	in	ADP
ajrhss-1265	80	4	medical	medical	ADJ
ajrhss-1265	80	5	terms	term	NOUN
ajrhss-1265	80	6	,	,	PUNCT
ajrhss-1265	80	7	they	they	PRON
ajrhss-1265	80	8	are	be	AUX
ajrhss-1265	80	9	used	use	VERB
ajrhss-1265	80	10	to	to	PART
ajrhss-1265	80	11	check	check	VERB
ajrhss-1265	80	12	the	the	DET
ajrhss-1265	80	13	growth	growth	NOUN
ajrhss-1265	80	14	of	of	ADP
ajrhss-1265	80	15	diseases	disease	NOUN
ajrhss-1265	80	16	in	in	ADP
ajrhss-1265	80	17	graphical	graphical	ADJ
ajrhss-1265	80	18	representation	representation	NOUN
ajrhss-1265	80	19	.	.	PUNCT
ajrhss-1265	81	1	differential	differential	ADJ
ajrhss-1265	81	2	equations	equation	NOUN
ajrhss-1265	81	3	are	be	AUX
ajrhss-1265	81	4	useful	useful	ADJ
ajrhss-1265	81	5	in	in	ADP
ajrhss-1265	81	6	describing	describe	VERB
ajrhss-1265	81	7	mathematical	mathematical	ADJ
ajrhss-1265	81	8	models	model	NOUN
ajrhss-1265	81	9	involving	involve	VERB
ajrhss-1265	81	10	population	population	NOUN
ajrhss-1265	81	11	growth	growth	NOUN
ajrhss-1265	81	12	or	or	CCONJ
ajrhss-1265	81	13	radioactive	radioactive	ADJ
ajrhss-1265	81	14	decay	decay	NOUN
ajrhss-1265	81	15	.	.	PUNCT
ajrhss-1265	82	1	conclusion	conclusion	NOUN
ajrhss-1265	82	2	an	an	DET
ajrhss-1265	82	3	equation	equation	NOUN
ajrhss-1265	82	4	that	that	PRON
ajrhss-1265	82	5	contains	contain	VERB
ajrhss-1265	82	6	the	the	DET
ajrhss-1265	82	7	derivative	derivative	NOUN
ajrhss-1265	82	8	of	of	ADP
ajrhss-1265	82	9	an	an	DET
ajrhss-1265	82	10	unknown	unknown	ADJ
ajrhss-1265	82	11	function	function	NOUN
ajrhss-1265	82	12	is	be	AUX
ajrhss-1265	82	13	called	call	VERB
ajrhss-1265	82	14	a	a	DET
ajrhss-1265	82	15	differential	differential	ADJ
ajrhss-1265	82	16	equation	equation	NOUN
ajrhss-1265	82	17	.	.	PUNCT
ajrhss-1265	83	1	the	the	DET
ajrhss-1265	83	2	rate	rate	NOUN
ajrhss-1265	83	3	of	of	ADP
ajrhss-1265	83	4	change	change	NOUN
ajrhss-1265	83	5	of	of	ADP
ajrhss-1265	83	6	a	a	DET
ajrhss-1265	83	7	function	function	NOUN
ajrhss-1265	83	8	at	at	ADP
ajrhss-1265	83	9	a	a	DET
ajrhss-1265	83	10	point	point	NOUN
ajrhss-1265	83	11	is	be	AUX
ajrhss-1265	83	12	defined	define	VERB
ajrhss-1265	83	13	by	by	ADP
ajrhss-1265	83	14	the	the	DET
ajrhss-1265	83	15	derivatives	derivative	NOUN
ajrhss-1265	83	16	of	of	ADP
ajrhss-1265	83	17	the	the	DET
ajrhss-1265	83	18	function	function	NOUN
ajrhss-1265	83	19	.	.	PUNCT
ajrhss-1265	84	1	a	a	DET
ajrhss-1265	84	2	differential	differential	ADJ
ajrhss-1265	84	3	equation	equation	NOUN
ajrhss-1265	84	4	relates	relate	VERB
ajrhss-1265	84	5	these	these	DET
ajrhss-1265	84	6	derivatives	derivative	NOUN
ajrhss-1265	84	7	with	with	ADP
ajrhss-1265	84	8	the	the	DET
ajrhss-1265	84	9	other	other	ADJ
ajrhss-1265	84	10	functions	function	NOUN
ajrhss-1265	84	11	.	.	PUNCT
ajrhss-1265	85	1	differential	differential	ADJ
ajrhss-1265	85	2	equations	equation	NOUN
ajrhss-1265	85	3	are	be	AUX
ajrhss-1265	85	4	mainly	mainly	ADV
ajrhss-1265	85	5	used	use	VERB
ajrhss-1265	85	6	in	in	ADP
ajrhss-1265	85	7	the	the	DET
ajrhss-1265	85	8	fields	field	NOUN
ajrhss-1265	85	9	of	of	ADP
ajrhss-1265	85	10	biology	biology	NOUN
ajrhss-1265	85	11	,	,	PUNCT
ajrhss-1265	85	12	physics	physics	NOUN
ajrhss-1265	85	13	,	,	PUNCT
ajrhss-1265	85	14	engineering	engineering	NOUN
ajrhss-1265	85	15	,	,	PUNCT
ajrhss-1265	85	16	and	and	CCONJ
ajrhss-1265	85	17	many	many	ADJ
ajrhss-1265	85	18	.	.	PUNCT
ajrhss-1265	86	1	the	the	DET
ajrhss-1265	86	2	main	main	ADJ
ajrhss-1265	86	3	purpose	purpose	NOUN
ajrhss-1265	86	4	of	of	ADP
ajrhss-1265	86	5	the	the	DET
ajrhss-1265	86	6	differential	differential	ADJ
ajrhss-1265	86	7	equation	equation	NOUN
ajrhss-1265	86	8	is	be	AUX
ajrhss-1265	86	9	for	for	ADP
ajrhss-1265	86	10	studying	study	VERB
ajrhss-1265	86	11	the	the	DET
ajrhss-1265	86	12	solutions	solution	NOUN
ajrhss-1265	86	13	that	that	PRON
ajrhss-1265	86	14	satisfy	satisfy	VERB
ajrhss-1265	86	15	the	the	DET
ajrhss-1265	86	16	equations	equation	NOUN
ajrhss-1265	86	17	and	and	CCONJ
ajrhss-1265	86	18	the	the	DET
ajrhss-1265	86	19	properties	property	NOUN
ajrhss-1265	86	20	of	of	ADP
ajrhss-1265	86	21	the	the	DET
ajrhss-1265	86	22	solutions	solution	NOUN
ajrhss-1265	86	23	.	.	PUNCT
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ajrhss-1265	87	2	us	we	PRON
ajrhss-1265	87	3	discuss	discuss	VERB
ajrhss-1265	87	4	the	the	DET
ajrhss-1265	87	5	definition	definition	NOUN
ajrhss-1265	87	6	,	,	PUNCT
ajrhss-1265	87	7	types	type	NOUN
ajrhss-1265	87	8	,	,	PUNCT
ajrhss-1265	87	9	methods	method	NOUN
ajrhss-1265	87	10	to	to	PART
ajrhss-1265	87	11	solve	solve	VERB
ajrhss-1265	87	12	the	the	DET
ajrhss-1265	87	13	differential	differential	ADJ
ajrhss-1265	87	14	equation	equation	NOUN
ajrhss-1265	87	15	,	,	PUNCT
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ajrhss-1265	87	17	,	,	PUNCT
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ajrhss-1265	87	20	of	of	ADP
ajrhss-1265	87	21	the	the	DET
ajrhss-1265	87	22	differential	differential	ADJ
ajrhss-1265	87	23	equation	equation	NOUN
ajrhss-1265	87	24	,	,	PUNCT
ajrhss-1265	87	25	types	type	NOUN
ajrhss-1265	87	26	of	of	ADP
ajrhss-1265	87	27	differential	differential	ADJ
ajrhss-1265	87	28	equations	equation	NOUN
ajrhss-1265	87	29	,	,	PUNCT
ajrhss-1265	87	30	with	with	ADP
ajrhss-1265	87	31	real	real	ADJ
ajrhss-1265	87	32	-	-	PUNCT
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ajrhss-1265	87	35	,	,	PUNCT
ajrhss-1265	87	36	and	and	CCONJ
ajrhss-1265	87	37	practice	practice	NOUN
ajrhss-1265	87	38	problems	problem	NOUN
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ajrhss-1265	88	5	2	2	X
ajrhss-1265	88	6	.	.	X
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ajrhss-1265	89	28	g	g	NOUN
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ajrhss-1265	90	6	-	-	PUNCT
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ajrhss-1265	90	8	-	-	PUNCT
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ajrhss-1265	90	11	order.html	order.html	NUM
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ajrhss-1265	92	2	/	/	SYM
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