Ratio Mathematica
Vol. 34, 2018, pp. 77–83

ISSN: 1592-7415
eISSN: 2282-8214

Notes on the Solutions of the First Order
Quasilinear Differential Equations

Alena Vagaská∗, Dušan Mamrilla†

Received: 11-05-2018. Accepted: 24-06-2018. Published: 30-06-2018

doi:10.23755/rm.v34i0.414

c©Dušan Mamrilla et al.

Abstract

The system of the quasilinear differential first order equations with the anti-
symetric matrix and the same element f (t, x (t)) on the main diagonal have
the property that r′ (t) = f (t, x (t)) r (t), where r (t) ≥ 0 is the polar func-
tion of the system. In special cases, when values f (t, x (t)) and g (t, x (t))
are only dependent on r2 (t), t ∈ J0 we can find the general solution of the
system (1) explicitly.
Keywords: nonlinear; quasilinear; differential equation; differential system;
2010 AMS subject classifications: 34C10.

∗Technical University of Košice, Faculty of Manufacturing Technologies, Prešov, Slovakia.
alena.vagaska@tuke.sk
†Prešov, Slovakia. dusan.mamrilla@gmail.com

77



A. Vagaská and D. Mamrilla

1 Introduction
Norkin, S. B. and Tchartorickij, J. A. [1] and Kurzweill, J. [2] investigated

the oscillatory properties of the 1,2-nontrivial solutions x(t) of systems of two
first order linear differential equations applying polar coordinates. Mamrilla, D.
and Norkin, S. B. [3] investigated the oscilatory properties of the 1,2,3-nontrivial
solutions x(t) of systems of three first order linear differential equations applying
spherical coordinates.
Applying polar (spherical) coordinates, the boundedness and oscillatority of the
1,2 (1,2,3)-nontrivial solutions x(t) of systems of two (three) first order quasi-
linear differential equations have been investigated by Mamrilla, D. [4], [5], [6]
and Mamrilla, D. and Seman, J. and Vagaská, A. [7], while special attention was
paid to the study of the properties of the x(t) solutions of the systems, the matrix
of which has the same element f (t,x(t)) on the main diagonal.

This paper gives some asymptotical and oscillatory properties of the solutions
to the system of the nonlinear differential equations:

 x1x2
x3




′

=


 f (t,x(t)) 0 g (t,x(t))0 f (t,x(t)) 0
−g (t,x(t)) 0 f (t,x(t))


 ·


 x1x2

x3


 , (1)

where t > 0, 0 6= f (t,x(t)), 0 6= g (t,x(t)) ∈ C0 (D ≡ J ×R3,R).
We assume that each solution

x(t) = (x1 (t) ,x2 (t) ,x3 (t)) , (2)
x1 (t0) = x

0
1,

x2 (t0) = x
0
2,

x3 (t0) = x
0
3, t0 ∈ J

exists on the interval J and we denote h > t0 > 0 the right endpoint of the interval
J and J0 = [t0,h) .

We shall denote

g1 (t,x) = f (t,x)x1 + g (t,x)x3,

g2 (t,x) = f (t,x)x2, (3)
g3 (t,x) = −g (t,x)x1 + f (t,x)x3.

It is known that if D0 ⊂ D is open nonempty set and derivatives (∂gi(t,x)/∂xj)
are continuous functions on D0 for every i,j ∈ {1,2,3} then each point
(t0,x

0
1,x

0
2,x

0
3) ∈ D0 is passed by one and only one integral curve x ∈ D of the

system (1) [3].

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Notes on the Solutions of the First Order Quasilinear Differential Equations

Definition 1.1. The solution x(t) to the system (1) is called i − trivial,
i ∈ {1,2,3} is fixed, if xi (t) = 0 on the interval J0. Otherwise x(t) is i −
nontrivial solution. If for at least one i ∈ {1,2,3} the solution to the system (1)
is i−nontrivial, shortly so solution x(t) is said to be nontrivial.

It is obvious that system (1) has 1,2,3 − trivial solution; 1,3 − trivial and
2 − nontrivial solution; 1,3 − nontrivial and 2 − trivial solution; 1,2,3 −
nontrivial solution.

Definition 1.2. The solution x(t) to the system (1) is called i − positive (i −
negative), i ∈ {1,2,3} is fixed, if xi (t) is positive (negative) function on the
interval J0.

Definition 1.3. The solution x(t) to the system (1) is called i − nondecreasing
(i−nonincreasing), i ∈{1,2,3} is fixed, if xi (t) is nondecreasing (nonincreas-
ing) function on the interval J0.

It is obvious that if f (t,x)x2 ≥ 0 (f (t,x)x2 ≤ 0) for any point (t,x) ∈
D then arbitrary solution x(t), t ∈ J0 to the system (1) is 2 − nondecreasing
(2−nonincreasing).

Definition 1.4. The solution x(t) to the system (1) is called i − bounded, i ∈
{1,2,3} is fixed, if xi (t) is the bounded function on interval J0. At other cases
x(t) is i − unbounded one which is called i − from above (i − from below)
unbounded, i ∈ {1,2,3} is fixed, if xi (t) is from above (from below) unbounded
function on interval J0.

It is obvious that if for every continuous function y defined on interval J0 :

a) sup
y

(∫ h
t0
|f (t,y)y2|dt

)
< ∞, then any solution x(t), t ∈ J0 to the system

(1) is 2− bounded,

b) sup
y

(∫ h
t0
f (t,y)y2dt

)
= −∞

(
inf
y

(∫ h
t0
f (t,y)y2dt

)
= ∞

)
then there ex-

ists a point t∗ ≥ t0 and 2−negative (2−positive) solution x(t), t ∈ [t∗,h)
to the system (1) such that it is 2 − from below (2 − from above) un-
bounded.

Definition 1.5. The solution x(t) to the system (1) is called i − oscillatory,
i ∈ {1,2,3} is fixed, if xi (t) is the oscillatory function, i. e. if there exists the
increasing sequence {tn}

∞
n=1 such that tn ∈ J0, tn → h and xi (tn) .xi (tn+1) < 0

for each n ∈ N. The solution x(t) is called i − nonoscillatory if there exists
h1 < h such that xi (t) is not changing its sign on the interval [h1,h), resp. if it
has maximally finite number of zero point on the interval [t0,h).

79



A. Vagaská and D. Mamrilla

2 Main results
Theorem 2.1. The general solution to the system (1) is generated by the trinity of
the functions:

x1 (t) =

(
C2 cos

(∫ t
t0

g (s,x(s))ds

)
−C3 sin

(∫ t
t0

g (s,x(s))ds

))

×exp


 t∫

t0

f (s,x(s))ds


 ,

x2 (t) = C1 exp

(∫ t
t0

f (s,x(s))ds

)
,

x3 (t) =

(
−C2 sin

(∫ t
t0

g (s,x(s))ds

)
−C3 cos

(∫ t
t0

g (s,x(s))ds

))
×exp

(∫ t
t0

f (s,x(s))ds

)
,

where Ci (i = 1,2,3) ∈ R are arbitrary constants.

Proof. The characteristic quasipolynomial of the system (1) is

det(A(t,x(t))−λ(t,x(t))E) =
= (f (t,x(t))−λ(t,x(t)))3 + g2 (t,x(t)) (f (t,x(t))−λ(t,x(t))) = 0

the solutions of which are the functions

λ1 (t,x(t)) = f (t,x(t)) and

λ2,3 (t,x(t)) = f (t,x(t))± ig (t,x(t)) .

The fundamental system of the solutions to the system (1) is generated by the
vector functions X1 (t,x(t)), ReXc2 (t,x(t)), ImX

c
2 (t,x(t)), where

X1 (t,x(t)) =


 01

0


exp(∫ t

t0

f (s,x(s))ds

)

Xc2 (t,x(t)) =




 10

0


 + i


 00
−1




exp(∫ t

t0

(f (s,x(s))− ig (s,x(s)))ds
)
,

80



Notes on the Solutions of the First Order Quasilinear Differential Equations

e.g.,

X2 (t,x(t)) =

=




 10

0


cos(∫ t

t0

g (s,x(s))ds

)
+


 00
−1


sin(∫ t

t0

g (s,x(s))ds

)
× exp

(∫ t
t0

f (s,x(s))ds

)
,

X3 (t,x(t)) =

=




 00
−1


cos(∫ t

t0

g (s,x(s))ds

)
−


 10

0


sin(∫ t

t0

g (s,x(s))ds

)
× exp

(∫ t
t0

f (s,x(s))ds

)
.

This proves the theorem.2

Corolary 2.1. If we put g (t,x(t)) = 1 in Theorem (2.1), we obtain assertion of
Theorem (2.1) in [7].

Theorem 2.2. Let for all continuous functions y defined on the interval J0:

a) sup
y

(∫ h
t0
|f (s,y) |ds

)
< ∞, then each solution x(t), t ∈ J0 to the system

(1) is 1,2,3− bounded,

b) sup
y

(∫ h
t0
f (s,y)ds

)
= −∞, then each solution x(t), t ∈ J0 to the system

(1) is 1,2,3 − bounded and such that x1 (t) → 0, x2 (t) → 0, x3 (t) → 0
for t → h,

c) inf
y

(∫ h
t0
f (s,y)ds

)
= ∞, then each solution x(t), t ∈ J0 to the system (1)

is such that it is i−unbounded at least for one i ∈{1,2,3} .

Proof. Theorem (2.1)implies that the general solution to the system (1) fulfils
a condition x21 (t) +x

2
2 (t) +x

2
3 (t) = (C

2
1 + C

2
2 + C

2
3) exp

(
2
∫ t
t0
f (s,x(s))ds

)
,

and this implies the assertion of the theorem. 2
We assume that for each nontrivial solution x(t), t ∈ J0 to the system (1)

there exists the trinity of the functions r (t) > 0, u(t), v (t) ∈ C1 (J0,R) such
that the coordinates xi (t), t ∈ J0, i = 1,2,3 fulfil [7]:

81



A. Vagaská and D. Mamrilla

x1 (t) = r (t) cosu(t) ,

x2 (t) = r (t) sinu(t) cosv (t) ,

x3 (t) = r (t) sinu(t) sinv (t) , (4)
r
′
(t) = x

′

1 (t) cosu(t) + x
′

2 (t) sinu(t) cosv (t) +

+ x
′

3 (t) sinu(t) sinv (t) ,

r (t)u′ (t) = −x
′

1 (t) sinu(t) + x
′

2 (t) cosu(t) cosv (t) +

+ x
′

3 (t) cosu(t) sinv (t) ,

r (t) sinu(t)v′ (t) = −x
′

2 (t) sinv (t) + x
′

3 (t) cosv (t) .

The function r (t) is called the polar, u(t) the first angle function and v (t)
the second angle function. From this after equivalent arrangement for nontrivial
solutions to the system (1) we get:

r′ (t) = f (t,x(t))r (t) ,

u′ (t) = −g (t,x(t)) sinv (t) , (5)
sinu(t)v′ (t) = −g (t,x(t)) cosu(t) cosv (t) .

3 Conclusions
The paper deals with qualitative and quantitative properties of the solutions

of special differential equations and systems of differential equations. Non-linear
and quasi-linear equations are less researched in mathematical publications, so the
goal of this paper was to investigate some asymptotical and oscillatory properties
of non-trivial solutions of such differential equations and systems thus contribut-
ing to knowledge in this field of research. Special attention was focused on the
study of the asymptotic and oscillatory properties of the x(t) solutions of the sys-
tems, the matrix of which has the same element on the main diagonal. We have
achieved new results due to the investigation of this subject by applying of polar
or spherical coordinates.

4 Acknowledgements
The research work is supported by the project KEGA 026TUKE-4/2016. Ti-

tle of the project: Implementation of Modern Information and Communication
Technologies in Education of Natural Science and Technical Subjects at Techni-
cal Faculties.

82



Notes on the Solutions of the First Order Quasilinear Differential Equations

References
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of a system of two linear differential equations by means of angle function,
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pp. 19-32

[2] J. Kurzweill, Ordinary differential equations, SNTL, Praha, 1978

[3] D. Mamrilla, S. B. Norkin, Investigation of oscillatory properties of linear
third order differential equations systems by means of angle functions, In-
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[4] D. Mamrilla On boundedness and oscillatoricity of certain differential equa-
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[5] D. Mamrilla The theory of angle functions and some properties of certain
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[6] D. Mamrilla On the systems of first order quasi - linear differential equa-
tions, Tribun EU Brno (2008)

[7] D. Mamrilla and J. Seman and A. Vagaská On the solutions of the first or-
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