Integrating Factor and First Integrals of a Class of Third Order ...

arXiv:1706.06199v1 [math.CA] 19 Jun 2017

Integrating Factor and First Integrals of a Class of Third Order Differential Equations M. Al-Jararha∗ Department of Mathematics, Yarmouk University, Irbid, Jordan, 21163.

Keywords and Phrases: Third order differential equation, Exact differential equations, None exact differential equations, Integrating factor, First integrals. AMS (2000) Subject Classification: 34A25, 34A30. Abstract The principle of finding an integrating factor for a none exact differential equations is extended to a class of third order differential equations. If the third order equation is not exact, under certain conditions, an integrating factor exists which transforms it to an exact one. Hence, it can be reduced into a second order differential equation. In this paper, we give explicit forms for certain integrating factors of a class of the third order differential equations.

1

Introduction

Third order nonlinear differential equations play a major role in Applied Mathematics, Physics, and Engineering [4, 8, 10, 14, 15]. To find the general solution of a third order nonlinear differential equation is not an easy problem in the general case. In fact, a very specific class of nonlinear third order differential equations can be solved by using special transformations. Other technique is to reduce the order of the differential equation into the second order, by finding a proper integrating factor. Recently, many studies appear to deal with the problem of the existence of an integrating factor for certain differential equations. For example, in [1, 3, 5, 9], the authors investigated the existence of an integrating factor of some classes of second order differential equations. In [3], the authors investigated the existence of an integrating factor of n − th order differential equations which has known symmetries of certain type. In [7], the authors improve some symbolic algorithms to compute the integrating factor for certain class of third order nonlinear differential equation. In this paper, we investigate the existence of an integrating factor of the following class of third order nonlinear differential equations: F3 (t, y, y ′ , y ′′ )y ′′′ + F2 (t, y, y ′ , y ′′ )y ′′ + F1 (t, y, y ′ , y ′′ )y ′ + F0 (t, y, y ′ , y ′′ ) = 0. ∗

[email protected]

1

(1.1)

In fact, we present some theoretical results related to the existence of certain forms of the integrating factor for (1.1). We also present some illustrative examples.

2

Integrating Factor and First Integral of a Class of Third Order Differential Equations

In this section, we investigate the existence of special forms of integrating factor of a class of none exact third order differential equations. In general, we said that the n − th order differ
ential equation f
t, y, y ′ , · · · , y (n−1) , y (n) = 0 is exact if
there exists a differentiable function
d ′ (n) = f t, y, y ′ , · · · , y (n−1) , y (n) = 0. Ψ t, y, y ′ , · · · , y (n) = c, such that dt Ψ t, y, y , · · · , y
In this case, Ψ t, y, y ′ , ·
· · , y (n) = c is called the first integral of the differential equation f t, y, y ′ , · · · , y (n−1) , y (n) = 0 (see for example [11, 13]). In [2], the author gave the conditions so that the first integral of     Fn t, y, y ′ , y ′′ , . . . , y (n−1) y (n) + Fn−1 t, y, y ′ , y ′′ , . . . , y (n−1) y (n−1) + · · ·     (2.1) + F1 t, y, y ′ , y ′′ , . . . , y (n−1) y ′ + F0 t, y, y ′ , y ′′ . . . , y (n−1) = 0.
exists. Also, he gave an explicit formula for Ψ t, y, y ′ , · · · , y (n) . for particular case, we consider the following class of third order differential equation: F3 (t, y, y ′ , y ′′ )y ′′′ + F2 (t, y, y ′ , y ′′ )y ′′ + F1 (t, y, y ′ , y ′′ )y ′ + F0 (t, y, y ′ , y ′′ ) = 0,

(2.2)

where F0 , F1 , F2 and F3 are continuous with their first partial derivatives with respect to t, y, y ′ , and y ′′ on some simply connected domain R in R4 . According to [2], this third order differential equation is exact if the following conditions: ∂y′′ F0 = ∂t F3 , ∂y′′ F1 = ∂y F3 , ∂y′′ F2 = ∂y′ F3 , ∂y′ F0 = ∂t F2 , ∂y′ F1 = ∂y F2 , and ∂y F0 = ∂t F1 (2.3) hold. Moreover, its first integral is given by Z y Z t


Ψ t, y, y ′ , y ′′ = F1 t0 , ξ, y ′ , y ′′ dξ F0 ξ, y, y ′ , y ′′ dξ + y0

t0 y′

+

Z

F2 t0 , y0 , ξ, y

y0′

= c,


′′

dξ +

Z

y ′′

y0′′


F3 t0 , y0 , y0′ , ξ dξ

(2.4)

for some constant c. Assume that (2.2) is not exact differential equation, then an integrating factor µ(t, y, y ′ , y ′′ ) if exists will transform equation it into an exact differential. According to the conditions (2.3), the integrating factor µ(t, y, y ′ , y ′′ ) of (2.2) must solve the following system of partial differential equations:   µ(y)F3t (y) + µt (y)F3 (y) = µ(y)F0y′′ (y) + µy′′ (y)F0 (y),     µ(y)F 2t (y) + µt (y)F2 (y) = µ(y)F0y ′ (y) + µy ′ (y)F0 (y),    µ(y)F1t (y) + µt (y)F1 (y) = µ(y)F0y (y) + µy (y)F0 (y), (2.5)  µ(y)F 1y ′ (y) + µy ′ (y)F1 (y) = µ(y)F2y (y) + µy (y)F2 (y),      µ(y)F1y′′ (y) + µy′′ (y)F1 (y) = µ(y)F2y (y) + µy (y)F2 (y),   µ(y)F2y′′ (y) + µy′′ (y)F2 (y) = µ(y)F3y′ (y) + µy′ (y)F3 (y), 2

where y = (t, y, y ′ , y ′′ ). To solve such system of partial differential equations is not easy. Therefore, to find an integrating factor of (2.2), we look for integrating factors of certain forms. Particularly, we are looking for an integrating factor of the form µ(ξ), where ξ := ξ(t, y, y ′ , y ′′ ) = α(t)β(y)γ(y ′ )δ(y ′′ ), where α(t), β(y) γ(y ′ ), and δ(y ′′ ) are differentiable functions. Substitute µ(ξ) = µ(α(t)β(y)γ(y ′ )δ(y ′′ )) in (2.5), then we get  µ(ξ)F3t (y) + µ′ (ξ)ξt F3 (y) = µ(ξ)F0y′′ (y) + µ′ (ξ)ξy′′ F0 (y),      µ(ξ)F2t (y) + µ′ (ξ)ξt F2 (y) = µ(ξ)F0y′ (y) + µ′ (ξ)ξy′ F0 (y),    µ(ξ)F1t (y) + µ′ (ξ)ξt F1 (y) = µ(ξ)F0y (y) + µ′ (ξ)ξy F0 (y), (2.6)  µ(ξ)F1y′ (y) + µ′ (ξ)ξy′ F1 (y) = µ(ξ)F2y (y) + µ′ (ξ)ξy F2 (y),     µ(ξ)F1y′′ (y) + µ′ (ξ)ξy′′ F1 (y) = µ(ξ)F2y (y) + µ′ (ξ)ξy F2 (y),    µ(ξ)F2y′′ (y) + µ′ (ξ)ξy′′ F2 (y) = µ(ξ)F3y′ (y) + µ′ (ξ)ξy′ F3 (y),

where µ′ (ξ) =

dµ dξ .

Equivalently, we have  ′ F3t (y) − F0y′′ (y) µ (ξ)   , =   µ(ξ) ξy′′ F0 (y) − ξt F3 (y)        µ′ (ξ) F1t (y) − F0y (y) = ,  µ(ξ) ξy F0 (y) − ξt F1 (y)        F1y′′ (y) − F2y (y)  µ′ (ξ)   , = µ(ξ) ξy F2 (y) − ξy′′ F1 (y)

F2t (y) − F0y′ (y) µ′ (ξ) , = µ(ξ) ξy′ F0 (y) − ξt F2 (y) F1y′ (y) − F2y (y) µ′ (ξ) , = µ(ξ) ξy F2 (y) − ξy′ F1 (y)

(2.7)

F2y′′ (y) − F3y′ (y) µ′ (ξ) . = µ(ξ) ξy′′ F2 (y) − ξy′ F3 (y)

Hence, an integrating factor µ(ξ) of (2.2) exists if a)

F3t (y)−F0y ′′ (y) F2t (y)−F0y ′ (y) F1y ′ (y)−F2y (y) F1y ′′ (y)−F2y (y) F1t (y)−F0y (y) ξy ′′ F0 (y)−ξt F3 (y) , ξy ′ F0 (y)−ξt F2 (y) , ξy F0 (y)−ξt F1 (y) , ξy F2 (y)−ξy ′ F1 (y) , ξy F2 (y)−ξy ′′ F1 (y) ,

and

F2y ′′ (y)−F3y ′ (y) ξy ′′ F2 (y)−ξy ′ F3 (y)

are functions of ξ,

and b) F3t (y) − F0y′′ (y) ξy′′ F0 (y) − ξt F3 (y) F1y′′ (y) − F2y (y) ξy F2 (y) − ξy′′ F1 (y)

= =

F2t (y) − F0y′ (y) F1y′ (y) − F2y (y) F1t (y) − F0y (y) = = = ′ ξy F0 (y) − ξt F2 (y) ξy F0 (y) − ξt F1 (y) ξy F2 (y) − ξy′ F1 (y) F2y′′ (y) − F3y′ (y) ξy′′ F2 (y) − ξy′ F3 (y)

hold. Therefore, we have the following theorem: Theorem 2.1. Let y = (t, y, y ′ , y ′′ ), and assume that Equation (2.2) is none exact differential equation. Then it admits an integrating factor µ(ξ) = µ(ξ(t, y, y ′ , y ′′ )) = µ (α(t)β(y)γ(y ′ )δ(y ′′ )), where α(t), β(y), γ(y ′ ), and δ(y ′′ ) are differentiable functions; if the conditions a)

F2t (y)−F0y ′ (y) F1y ′ (y)−F2y (y) F1y ′′ (y)−F2y (y) F3t (y)−F0y ′′ (y) F1t (y)−F0y (y) ξy ′′ F0 (y)−ξt F3 (y) , ξy ′ F0 (y)−ξt F2 (y) , ξy F0 (y)−ξt F1 (y) , ξy F2 (y)−ξy ′ F1 (y) , ξy F2 (y)−ξy ′′ F1 (y) ,

and

F2y ′′ (y)−F3y ′ (y) ξy ′′ F2 (y)−ξy ′ F3 (y)

are functions in ξ := α(t)β(y)γ(y ′ )δ(y ′′ ),

and 3

b) F3t (y) − F0y′′ (y) ξy′′ F0 (y) − ξt F3 (y) F1y′′ (y) − F2y (y) ξy F2 (y) − ξy′′ F1 (y)

= =

F2t (y) − F0y′ (y) F1y′ (y) − F2y (y) F1t (y) − F0y (y) = = = ξy′ F0 (y) − ξt F2 (y) ξy F0 (y) − ξt F1 (y) ξy F2 (y) − ξy′ F1 (y) F2y′′ (y) − F3y′ (y) , ξy′′ F2 (y) − ξy′ F3 (y)

hold. Moreover, the integrating factor is given by the formula  Z F3t (y) − F0y′′ (y) dξ . µ(ξ) = exp ξy′′ F0 (y) − ξt F3 (y) In the following sections, we presents some special cases of the above theorem. Moreover, we present some examples to illustrate the methodology of finding such integrating factor.

2.1

Integrating Factors of the Forms µ(α(t)), µ(β(y)), µ(γ(y ′)) and µ(δ(y ′′))

In this section, we give conditions so that an integrating factor of one of the forms µ(α(t)), µ(β(y)), µ(γ(y ′ )) and µ(δ(y ′′ )) for equation (2.2) exists. As a result of Theorem 2.1, we have the following corollaries: Corollary 2.1. Let y = (t, y, y ′ , y ′′ ), and assume that Equation (2.2) is none exact differential equation. Then it admits an integrating factor µ(ξ) = µ(α(t)), where α(t) is differentiable function; if the following two conditions hold: a) F1y′ (y) = F2y (y), F1y′′ (y) = F2y (y), and F2y′′ (y) = F3y′ (y), and b)

F0y ′′ (y)−F3t (y) ξt F3 (y)

=

F0y ′ (y)−F2t (y) ξt F2 (y)

=

F0y (y)−F1t (y) , ξt F1 (y)

and they are functions in ξ := ξ(α(t)).

Moreover, the integrating factor is given by  Z F0y′′ (y) − F3t (y) dξ . µ(ξ) = exp ξt F3 (y) Corollary 2.2. Let y = (t, y, y ′ , y ′′ ), and assume that Equation (2.2) is none exact differential equation. Then it admits an integrating factor µ(ξ) = µ(β(y)), where β(y) is differentiable function; if the following two conditions hold: a) F3t (y) = F0y′′ (y), F2t (y) = F0y′ (y), and F2y′′ (y) = F3y′ (y), and b)

F1t (y)−F0y (y) ξy F0 (y)

=

F1y ′ (y)−F2y (y) ξy F2 (y)

=

F1y ′′ (y)−F2y (y) , ξy F2 (y)

and they are functions in ξ := ξ(β(y)).

Moreover, the integrating factor is given by  Z F1t (y) − F0y (y) dξ . µ(ξ) = exp ξy F0 (y) 4

Corollary 2.3. Let y = (t, y, y ′ , y ′′ ), and assume that Equation (2.2) is none exact differential equation. Then it admits an integrating factor µ(ξ) = µ(γ(y ′ )) for some differentiable functions γ(y ′ ); if the two conditions hold: a) F3t (y) = F0y′′ (y), F1t (y) = F0y (y), and F1y′′ (y) = F2y (y), and b)

F2t (y)−F0y ′ (y) ξy ′ F0 (y)

=

F2y (y)−F1y ′ (y) ξy ′ F1 (y)

=

F3y ′ (y)−F2y ′′ (y) , ξy ′ F3 (y)

and they are functions in ξ := γ(y ′ ).

Moreover, the integrating factor is given by  Z F2t (y) − F0y′ (y) dξ . µ(ξ) = exp ξy′ F0 (y) Corollary 2.4. Let y = (t, y, y ′ , y ′′ ), and assume that Equation (2.2) is none exact differential equation. Then it admits an integrating factor µ(ξ) = µ(δ(y ′′ )) for some differentiable function δ(y ′′ ); if the following two conditions hold: a) F2t (y) = F0y′ (y), F1t (y) = F0y (y), and F1y′ (y) = F2y (y), and b)

F3t (y)−F0y ′′ (y) ξy ′′ F0 (y)

=

F2y (y)−F1y ′′ (y) ξy ′′ F1 (y)

=

F2y ′′ (y)−F3y ′ (y) , ξy ′′ F2 (y)

and they are functions in ξ := δ(y ′′ ).

Moreover, the integrating factor is given by Z  F3t (y) − F0y′′ (y) µ(ξ) = exp dξ . ξy′′ F0 (y) − ξt F3 (y) Example 2.1. Consider the differential equation (y ′ )3 y ′′′ + 2yy ′′ − (y ′ )2 + (y ′ )3 = 0.

(2.8)

Clearly, F3 (t, y, y ′ , y ′′ ) = (y ′ )3 , F2 (t, y, y ′ , y ′′ ) = 2y, F1 (t, y, y ′ , y ′′ ) = −y ′ , and F0 (t, y, y ′ , y ′′ ) = (y ′ )3 . Moreover, F3t (t, y, y ′ , y ′′ ) = F0y′′ (t, y, y ′ , y ′′ ) = 0, F1t (t, y, y ′ , y ′′ ) = F0y (t, y, y ′ , y ′′ ) = 0, and F2t (y)−F

′ (y)

F2y (y)−F

′ (y)

F

′ (y)−F

′′ (y)

1y 2y 0y = = 3y ξ ′ F3 (y) = −3 F1y′′ (t, y, y ′ , y ′′ ) = F2y (t, y, y ′ , y ′′ ) = 0, and y′ . ξy ′ F0 (y) ξy ′ F1 (y) y Therefor, the condition in Corollary 2.3 hold, and so an integrating factor of this third order differential equation exists and is given by the formula µ(y ′ ) = (y ′ )−3 . By multiplying (2.8) by µ(y ′ ) = (y ′ )−3 , we get
−3 ′′ y ′′′ + 2y y ′ y − (y ′ )−2 y ′ + 1 = 0. (2.9)

For this equation F3 (t, y, y ′ , y ′′ ) = 1, F2 (t, y, y ′ , y ′′ ) = 2y (y ′ )−3 , F1 (t, y, y ′ , y ′′ ) = −(y ′ )−2 , and F0 (t, y, y ′ , y ′′ ) = 1. Clearly, ∂y′′ F0 = ∂t F3 = 0, ∂y′′ F1 = ∂y F3 = 0, ∂y′′ F2 = ∂y′ F3 = 0, ∂y′ F0 = ∂t F2 = 0, ∂y′ F1 = ∂y F2 = 2(y ′ )−3 , and ∂y F0 = ∂t F1 = 0. Hence, it is exact differential equation, and its first integral is given by Z t Z y Z y′′ Z y′
Ψ t, y, y ′ , y ′′ = dξ − (y ′ )−2 dξ + 2y0 dξ = c, (2.10) ξ −3 dξ + t0

y0

y0′

y0′′

More precisely,
Ψ t, y, y ′ , y ′′ = y ′′ − (y ′ )−2 y + t = c 5

(2.11)

Example 2.2. consider the third order linear differential equation p2 (t)y ′′′ + αp2 (t)y ′′ + p1 (t)y ′ + p0 (t)y = h(t), p2 (t) 6= 0.

(2.12)

Assume that p2 (t)p′1 (t) − p′2 (t)p1 (t) = p0 (t)p2 (t). Then, by applying the conditions in Corollary 2.1, the above third order linear differential equation admits an integrating factor µ(t) = p21(t) . In fact, if we multiply Eq. (2.12) by µ(t) = p21(t) , then we get y ′′′ + αy ′′ +

h(t) p1 (t) ′ p0 (t) y + y= . p2 (t) p2 (t) p2 (t)

′  = From the condition p2 (t)p′1 (t) − p′2 (t)p1 (t) = p0 (t)p2 (t), we have pp12 (t) (t) above equation becomes     p1 (t) h(t) p1 (t) ′ ′′′ ′′ ′ y + αy + y= . y + p2 (t) p2 (t) p2 (t)

p0 (t) p2 (t) .

Hence, the

This equation can be written as     p1 (t) h(t) d . y ′′ + αy ′ + y = dt p2 (t) p2 (t) Hence, the first integral of Eq. (2.12) is given by   Z p1 (t) ′′ ′ y + αy + y= p2 (t)

t

h(s) + c. p2 (s)ds

Integrating Factors of the Forms µ(α(t)β(y)), µ(α(t)γ(y ′)), µ(α(t)δ(y ′′ )), µ(β(y)γ(y ′)), µ(β(y)δ(y ′′)) and µ(γ(y ′)δ(y ′′ ))

3

In this section, we give conditions so that an integrating factor of one of the forms µ(α(t)β(y)), µ(α(t)γ(y ′ )), µ(α(t)δ(y ′′ )), µ(β(y)γ(y ′ )), µ(β(y)δ(y ′′ )) and µ(γ(y ′ )δ(y ′′ )) for equation (2.2) exists. As a result of Theorem 2.1, we have the following corollaries: Corollary 3.1. Let y = (t, y, y ′ , y ′′ ), and assume that Equation (2.2) is none exact differential equation. Then it admits an integrating factor µ(ξ) = µ(α(t)β(y)), where α(t) and β(y)) are differentiable functions; if the following two conditions hold: a) F2y′′ (y) = F3y′ (y), and b)

F0y ′′ (y)−F3t (y) ξt F3 (y)

F

′ (y)−F2t (y)

F

′ (y)−F2y (y)

= 0y ξt F2 (y) = 1y ξy F2 (y) are functions in ξ := ξ(α(t)β(y)).

=

F1y ′′ (y)−F2y (y) ξy F2 (y)

Moreover, the integrating factor is given by  Z F0y′′ (y) − F3t (y) dξ . µ(ξ) = exp ξt F3 (y) 6

=

F1t (y)−F0y (y) ξy F0 (y)−ξt F1 (y) ,

and they

Corollary 3.2. Let y = (t, y, y ′ , y ′′ ), and assume that Equation (2.2) is none exact differential equation. Then it admits an integrating factor µ(ξ) = µ(α(t)γ(y ′ )) for some differentiable functions α(t) and γ(y ′ ); if the two conditions hold: a) F1y′′ (y) = F2y (y), and b)

F0y ′′ (y)−F3t (y) ξt F3 (y)

=

F0y (y)−F1t (y) ξt F1 (y)

=

F2y (y)−F1y ′ (y) ξy ′ F1 (y)

=

F3y ′ (y)−F2y ′′ (y) ξy ′ F3 (y)

=

F2t (y)−F0y ′ (y) ξy ′ F0 (y)−ξt F2 (y) .

and they

are functions in ξ := α(t)γ(y ′ ). Moreover, the integrating factor is given by  Z F0y′′ (y) − F3t (y) dξ . µ(ξ) = exp ξt F3 (y) Corollary 3.3. Let y = (t, y, y ′ , y ′′ ), and assume that Equation (2.2) is none exact differential equation. Then it admits an integrating factor µ(ξ) = µ(α(t)δ(y ′′ )) for some differentiable function δ(y ′′ ); if the following two conditions hold: a) F1y′ (y) = F2y (y), and b)

F0y ′ (y)−F2t (y) ξt F2 (y)

=

F0y (y)−F1t (y) ξt F1 (y)

are functions in ξ :=

=

F2y (y)−F1y ′′ (y) ξy ′′ F1 (y)

=

F2y ′′ (y)−F3y ′ (y) ξy ′′ F2 (y)

=

F3t (y)−F0y ′′ (y) . ξy ′′ F0 (y)

and they

α(t)δ(y ′′ ).

Moreover, the integrating factor is given by  Z F0y′ (y) − F2t (y) dξ . µ(ξ) = exp ξt F2 (y) Corollary 3.4. Let y = (t, y, y ′ , y ′′ ), and assume that Equation (2.2) is none exact differential equation. Then it admits an integrating factor µ(ξ) = µ(β(y)γ(y ′ )), where β(y) and γ(y ′ ) are differentiable functions; if the following two conditions hold: a) F3t (y) = F0y′′ and b)

F2t (y)−F0y ′ (y) ξy ′ F0 (y)

=

F1t (y)−F0y (y) ξy F0 (y)

are functions in ξ :=

=

F1y ′′ (y)−F2y (y) ξy F2 (y)

=

F3y ′ (y)−F2y ′′ (y) ξy ′ F3 (y)

=

F1y ′ (y)−F2y (y) ξy F2 (y)−ξy ′ F1 (y)

and they

β(y)γ(y ′ ).

Moreover, the integrating factor is given by  Z F2t (y) − F0y′ (y) dξ . µ(ξ) = exp ξy′ F0 (y) Corollary 3.5. Let y = (t, y, y ′ , y ′′ ), and assume that Equation (2.2) is none exact differential equation. Then it admits an integrating factor µ(ξ) = µ(β(y)δ(y ′′ )), where β(y) and δ(y ′′ ) are differentiable functions; if the following two conditions hold: a) F2t (y) = F0y′ (y), and 7

b)

F3t (y)−F0y ′′ (y) ξy ′′ F0 (y)

=

F1t (y)−F0y (y) ξy F0 (y)

are functions in ξ :=

=

F1y ′ (y)−F2y (y) ξy F2 (y)

=

F2y ′′ (y)−F3y ′ (y) ξy ′′ F2 (y)

=

F1y ′′ (y)−F2y (y) ξy F2 (y)−ξy ′′ F1 (y)

and they

β(y)δ(y ′′ ).

Moreover, the integrating factor is given by  Z F3t (y) − F0y′′ (y) dξ . µ(ξ) = exp ξy′′ F0 (y) Corollary 3.6. Let y = (t, y, y ′ , y ′′ ), and assume that Equation (2.2) is none exact differential equation. Then it admits an integrating factor µ(ξ) = µ(γ(y ′ )δ(y ′′ )), where γ(y ′ ) and δ(y ′′ ) are differentiable functions; if the following two conditions hold: a) F1t (y) = F0y (y), and b)

F3t (y)−F0y ′′ (y) ξy ′′ F0 (y)

they are

F2t (y)−F0y ′ (y) F2y (y)−F1y ′ (y) = ξy ′ F0 (y) ξy ′ F1 (y) functions in ξ = γ(y ′ )δ(y ′′ ).

=

=

F2y (y)−F1y ′′ (y) ξy ′′ F1 (y)

=

F2y ′′ (y)−F3y ′ (y) ξy ′′ F2 (y)−ξy ′ F3 (y)

and

Moreover, the integrating factor is given by  Z F3t (y) − F0y′′ (y) dξ . µ(ξ) = exp ξy′′ F0 (y)

4

Integrating Factors of the Forms µ(α(t)β(y)γ(y ′)), µ(α(t)β(y)δ(y ′′)), µ(α(t)γ(y ′)δ(y ′′)) and µ(β(y)γ(y ′)δ(y ′′ ))

In this section, we give conditions so that an integrating factor of one of the forms µ(α(t)β(y)γ(y ′ )), µ(α(t)β(y)δ(y ′′ )), µ(α(t)γ(y ′ )δ(y ′′ )) and µ(β(y)γ(y ′ )δ(y ′′ )) for equation (2.2) exists. As a result of Theorem 2.1, we have the following corollaries: Corollary 4.1. Let y = (t, y, y ′ , y ′′ ), and assume that Equation (2.2) is none exact differential equation. Then it admits an integrating factor µ(ξ) = µ(ξ(t, y, y ′ )) = µ (α(t)β(y)γ(y ′ )) where α(t), β(y), and γ(y ′ ) are differentiable functions; if F0y′′ (y) − F3t (y) ξt F3 (y) F1t (y) − F0y (y) ξy F0 (y) − ξt F1 (y)

= =

F2t (y) − F0y′ (y) F1y′′ (y) − F2y (y) F3y′ (y) − F2y′′ (y) = = = ξy F2 (y) ξy′ F3 (y) ξy′ F0 (y) − ξt F2 (y) F1y′ (y) − F2y (y) ξy F2 (y) − ξy′ F1 (y)

and they are functions in ξ := α(t)β(y)γ(y ′ ). Moreover, the integrating factor is given by the formula  Z F0y′′ (y) − F3t (y) dξ . µ(ξ) = exp ξt F3 (y) Corollary 4.2. Let y = (t, y, y ′ , y ′′ ), and assume that Equation (2.2) is none exact differential equation. Then it admits an integrating factor µ(ξ) = µ(α(t)β(y)δ(y ′′ )) where α(t), β(y), and δ(y ′′ ) are differentiable functions; if F0y′ (y) − F2t (y) ξt F2 (y) F3t (y) − F0y′′ (y) ξy′′ F0 (y) − ξt F3 (y)

= =

F1y′ (y) − F2y (y) F2y′′ (y) − F3y′ (y) F1y′′ (y) − F2y (y) = = = ξy′′ F2 (y) ξy F2 (y) ξy F2 (y) − ξy′′ F1 (y) F1t (y) − F0y (y) ξy F0 (y) − ξt F1 (y) 8

and they are functions in ξ = α(t)β(y)δ(y ′′ ). Moreover, the integrating factor is given by Z  F0y′ (y) − F2t (y) µ(ξ) = exp dξ . ξt F2 (y) Corollary 4.3. Let y = (t, y, y ′ , y ′′ ), and assume that Equation (2.2) is none exact differential equation. Then it admits an integrating factor µ(ξ) = µ(α(t)γ(y ′ )δ(y ′′ )) where α(t), γ(y ′ ), and δ(y ′′ ) are differentiable functions; if F0y (y) − F1t (y) ξt F1 (y) F2t (y) − F0y′ (y) ξy′ F0 (y) − ξt F2 (y)

= =

F2y (y) − F1y′′ (y) F3t (y) − F0y′′ (y) F2y (y) − F1y′ (y) = = ξy′ F1 (y) ξy′′ F1 (y) ξy′′ F0 (y) − ξt F3 (y) F2y′′ (y) − F3y′ (y) ξy′′ F2 (y) − ξy′ F3 (y)

and they are functions in ξ = α(t)γ(y ′ )δ(y ′′ ). Moreover, the integrating factor is given by  Z F0y (y) − F1t (y) dξ . µ(ξ) = exp ξt F1 (y) Corollary 4.4. Let y = (t, y, y ′ , y ′′ ), and assume that Equation (2.2) is none exact differential equation. Then it admits an integrating factor µ(ξ) = µ(β(y)γ(y ′ )δ(y ′′ )), where β(y), γ(y ′ ) and δ(y ′′ ) are differentiable functions; if F3t (y) − F0y′′ (y) ξy′′ F0 (y) F1y′′ (y) − F2y (y) ξy F2 (y) − ξy′′ F1 (y)

= =

F2t (y) − F0y′ (y) F1y′ (y) − F2y (y) F1t (y) − F0y (y) = = = ξy′ F0 (y) ξy F0 (y) ξy F2 (y) − ξy′ F1 (y) F2y′′ (y) − F3y′ (y) ξy′′ F2 (y) − ξy′ F3 (y)

and they are functions in ξ := β(y)γ(y ′ )δ(y ′′ ). Moreover, the integrating factor is given by Z  F3t (y) − F0y′′ (y) µ(ξ) = exp dξ . ξy′′ F0 (y)

5

Concluding Remarks

In this paper, we investigated the existence of integrating factor of the following class of third order nonlinear differential equations: F3 (t, y, y ′ , y ′′ )y ′′′ + F2 (t, y, y ′ , y ′′ )y ′′ + F1 (t, y, y ′ , y ′′ )y ′ + F0 (t, y, y ′ , y ′′ ) = 0.

(5.1)

In fact, we presented some theoretical results related to the existence of certain forms of an integrating factor of (5.1). Also, we presented some illustrative examples. In fact, these results not only useful for finding an integrating factor of (5.1) analytically but also computationally. Particularly, we can check the validity of the conditions in our results by using the symbolic toolboxes in MATLAB and MAPLE softwares. Also, we can find the integrating factor using the formula in these results with the help of the symbolic toolboxes. Moreover, by using the same argument above, we can derive an integrating factor in terms of ξ = α(t) + β(y) + γ(y ′ ) + δ(y ′′ ). 9

Acknowledgment We would like to thank the editor and the referees for their valuable comments on this paper.

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