Special Multiplicative Operators for the Solution of ODE – Invariants and Representations

Special Multiplicative Operators for the Solution of ODE –

Invariants and Representations

Zenonas Navickas

1

, Tadas Telksnys

2

, Minvydas Ragulskis

2,*

1_{Department of Applied Mathematics, Kaunas University of Technology, Kaunas, Lithuania.}

2_{Research Group for Mathematical and Numerical Analysis of Dynamical Systems, Kaunas University of Technology, Kaunas,} Lithuania.

Received 12 July 2014; received in revised form 05 August 2014; accepted 10 September 2014

Abstract

The generalized multiplicative operator of differentiation is introduced in this paper. It is shown that the

generalized multiplicative operator can be expressed as a product of two noncommutative but multiplicative

exponential operators, though the generalized multiplicative operator is not an exponential operator itself. The

generalized multiplicative operator is effectively exploited for the construction of solutions to nonlinear ordinary

differential equations through formal transformations of invariants and representations of initial conditions. The

concept of the generalized multiplicative operator provides the insight into the algebraic structure of solutions to

nonlinear ordinary differential equations which cannot be identified using conventional exponential operators.

Keywords: ordinary differential equation, multiplicative operator, invariant

1.

Introduction

The construction of analytic solutions to nonlinear ordinary differential equations (ODE) is an important research topic.

Numerous techniques have been developed for that purpose during the last decades. An explicit algorithm based on the Laurent

series for the construction of meromorphic solutions of autonomous nonlinear ODE is presented in [1]. The frequency domain

approach is used to prove the existence of a unique bounded, exponentially stable solution to some third order nonlinear

differential equations [2]. Existence and boundary behavior for singular nonlinear ODE is investigated in [3]. WTC-Kruskal

algorithm is developed in [4] in order to study the Painleve property of nonlinear ODE. Differential transform method has been

successfully exploited for solving nonlinear ODE and their systems [5, 6]. The Adomian decomposition method is used to

construct the solution in a form of an infinite series where the components are usually determined recurrently [7].

Semi-analytical Chebyshev collocation method is used to solve high-order nonlinear ODE in [8]. We list only a small fraction of

different techniques and semi-analytical algorithms for the construction of exact solutions to ODE.

The application of algebraic techniques for the construction of analytic solutions to ODE is a classical field of research [9,

10]. An overview on developments of algebraic theory approach to ODE is given in [11]. The application of algebraic theory to

the numerical treatment of ODE is studied in [12]. The differential operator is one of the key concepts in the algebraic theory of

differential equations [13]. The exponential differential operator is especially useful for these purposes [14, 15].

*Corresponding author. E-mail address: [email protected]

It is well known that the concept of the invariant plays an important role in mathematics in general [16]. In particular,

special invariants of ODE have been recently formulated in the context of geometrical analysis of differential equations [17, 18].

The main objective of this paper is to introduce the concept of the generalized multiplicative operator of differentiation and to

demonstrate its applicability in solving practical problems. Moreover, we will demonstrate the relationship among the

generalized multiplicative operator of differentiation and invariants of differential equations what will help to develop special

techniques for the construction of analytic solutions to nonlinear ODE problems.

This paper is organized as follows. Symbols and notations are listed in section 2; the generalized multiplicative operator

is introduced in section 3; the expression of the solution in the operator form is derived in section 4; a number of examples are

given in section 5 and concluding remarks are given in the last section.

2.

Symbols and notations

The following notations will be used throughout the manuscript (appropriate definitions will be given later):

n – the order of the explicit ordinary differential equation; y – the dependent variable;

1 1 0,s, ,sn

s  – Cauchy parameters (initial conditions);

 – the canonical variable (the center of the series expansion of the solution); x – the free variable;



,s0,s1, ,sn1



p  – an R-valued function of  and Cauchy parameters;



x, ,s0,s1, ,sn1



f   – an R-valued function of x,  and Cauchy parameters;

1 1 0, , ,

, 



n

s s s 

 – the set of functions p



,s0,s1,,sn1



;

1 1 0, , ,

,

, _



n

s s s

x  – the set of functions f



x,,s0,s1,,sn1



;

s, t – Cauchy parameters for n2 (i.e. s:s₀; t:s₁); x

D , D_,

0

s

D , …, Ds_n1– ordinary differential operators in respect of variables x, , s0, …, sn1; y

D – the generalized differential operator;

M,M0 – multiplicative operators;

G – the generalized multiplicative operator;



,s0,s1, ,sn1



v  – the invariant associated with D_y;

3.

The generalized multiplicative operator

3.1 Existing designs (or original designs)

R

      

1 0

:

n

s s

j

p _  ; where _{   R}

1 0, ,

, s  s_n

 are variation intervals (or unions of intervals) of variables

R



1 0

,

,

,

s



s

n



. These functions are differentiable any number times in respect of every variable. It can be noted that the identification of variation intervals (or unions of intervals) is a straightforward task whenever the expression of



, 0, , n1



j s s

p   is given explicitly. For example, the function





_

_

0 0

4 1 1

1 ,

s s

p

  

 

(1)

is defined and differentiable any number of times in respect of  and s₀ when 



;0

 

 0;



and 

      

4 1 ;

0

s (the

principal square root is considered in Eq. (1)). The analysis of functions of the first type is not the objective of this paper, but we

will consider only such functions pj pj



,s₀,,sn₁



that intervals ,s₀,,sn₁ exist and are not empty sets. The set of functions of the first type is denoted as

1 1 0, , ,

, 



n

s s s 

 .

Functions of the second type are constructed from functions of the first type using the following algorithm.

(i) Construct the power series:





_







 





0

1 0

1 0

0

,

,

,

!

,

,

,

,

j

n j

j

n

p

s

s

j

x

s

s

x

f









(2)

(ii) Extend the function f₀



x,,s₀,,sn₁



to a wider domain (if it is possible) using classical extension techniques. The

extended function f



x,,s₀,,sn_1



is denoted as the second type function. For example, the series





_

_

_

_



_

_

 



 

   

  

  

     

   

    

  

  

0 0

0 0

0 0

4 1 1 4

1 1

1 !

! ,

,

j

j

j

j j

s x s

j j x s

x f

 



can be extended to a function





_



_



x s

s s

x f

  

  

0 0 0

4 1 1

4 1 1 ,

,

  

for s0



;0



and 



1 14s0



x. From now on we will use the equality









s





x

s

s j

j x

j

j j

  

        

   

    

  

 





 0

0

0 0 1 1 4

4 1 1

4 1 1

1 !

! 

 

assuming that the transformation into the extended function does not cause any misunderstandings and will not specify the domain of x,  and s₀.

Other forms of the second type functions can be used. Typical cases (structures of f0) are listed below:





_















  

0

1 0

1 0

0 , , ,

! ,

, , ,

j

n j

j

n p s s

j x s

s x





_











 

0

1 0

1 0

0 , , ,

! ,

, ,

j

n j

j

n p s s

j s

s

f      .

It can be noted that it is not necessary to introduce the function norm (neither for the first type functions nor for the second

type functions) in the process of the construction of analytic solutions of nonlinear ordinary differential equations.

The set of extended functions is denoted as

1 0, ,

,

, 



n

s s

x  (,s0,,sn1x,,s0,,sn1).

3.2 The generalized operator of differentiation Let us consider an explicit ODE:

    

  

 , , , , _n1_1 n n

n n

dx y d dx dy y x P dx

y d

 ; (3)

with initial conditions





k x k

n k

s dx

s s x y

d _

 



, 0, , 1

, 

;k0,1,,n1; (4)

where yy



x,,s₀,,sn₁



is the solution to the initial value problem (Eq. (3, 4)); nN is the fixed order of the differential

equation; PnPn



,s₀,,sn_1



is a function of the first type. Then, the generalized operator of differentiation Dy associated with Eq. (1) reads [19]:





₁

2 1

0 2 1 0 1

1 , , ,

:     _{ }  _{ }

n n n n s

s n s

s

y D sD s D s D P s s D

D _    (5)

This paper is organized as follows. Symbols and notations are enlisted in section 2; the generalized multiplicative

operator is introduced in section 3; the expression of the solution in the operator form is derived in section 4; a number of

examples are given in section 5 and concluding remarks are given in the last section.

Conventional properties of differentiation hold for D_y. Several properties are listed below:

(i) Dy



c1f1c2f2



c1Dyf1c2Dyf2; where

c

1

,

c

2



R

;

f

1

,

f

2





x,,s0,,sn1.

(ii) Dy



f₁f₂







Dyf₁



f₂ f₁



Dyf₂



.

(iii)

 

 

 

2 2

2 1 2 1 2 1

f Df f f Df f f

D_y   .

(iv)







  







        

m j

j m y j y m

y D f D f

j m f

f D

0

2 1

2

1 ; where





   

 

 

     

; , ! !

! ; , 0

m j j m j

m m j

j m

 , 2 , 1 , 0 ,m

j .

(v) Let f

 

z be a function differentiable any number of times in respect of the variable z. If



x, ,s0, ,sn1



:f



f1



x, ,s0, ,sn1





F     then



 



_{ , , , , } 1



, , 0, , 1



1 0

1 

 



 y n

s s x f z z

yF D f z D f x s s

D

n



 

 .

Proof.

Without the loss of generality we will prove Property (ii) for n2. Let us assume that the generalized operator of differentiation reads D_y P

 

s,t D_s Q

 

s,t D_t and functions PP

 

s,t ; QQ

 

s,t ; f1f1

 

s,t ; f2 f2

 

s,t are

differentiable in respect of variables s and t any number of times. Then,



 







































1 2 1 2





1



2



1



2







1



2 1



2



.

2 1 2 1 2

1 2 1 2

1 2

1

f D f f f D f D Q f f D P f f f D Q f f D P

f D f f f D Q f D f f f D P f f QD PD f

f D

y y

t s

t s

t t

s s

t s y

 

 

 

 

 

  

 

End of proof.

Other properties can be proved analogously.

3.3 The multiplicative operator y

D can be exploited to construct the multiplicative operator M:









0 !

: j

j y j

D j x

M . (6)

The operator M satisfies following equalities [19, 20]:

(i) M



a₁f₁a₂f₂



a₁Mf₁a₂Mf₂; a₁,a₂R;

1 0, ,

, 2

1,f  s sn

f _{ } .

(ii) Mm 



x



m; mZ0.

(iii) Mf₁



,s₀,,sn₁



 f₁



x,Ms₀,,Msn₁



.

(iv)



_



_



_



_

1 0

2

1 0

1 1 0

2

1 0

1

, , ,

, , , ,

, ,

, , ,

 

 

  

n n n

n

Ms Ms

x f

Ms Ms

x f s s f

s s f M

  



  



.

Without the loss of generality we will prove the equality Mf1

 

s,t  f1



Ms,Mt



when DyP

 

s,t Ds Q

 

s,t Dt. Proof.

Let y1:Msy1



x,s,t



; y2:Mty2



x,s,t



; z:Mf1

  

s,t z x,s,t



and w: f1



Ms,Mt



 f1



y1



x,s,t

 

,y2 x,s,t





.

Then,

 

 

_

_

 

 

 

 

,

 

, .

, ,

! ,

! 1 ,

! ,

1 1

1 0

1 1

1 1 0

1 1

t s Mf QD t s Mf PD

t s Mf D t s f D j x D t s f D j x t

s f D j x x t s Mf D

t s

y j

j y j

y j

j y j

j

j y j

x

 

 

 

     

   













 



  





The last equality yields the following differential equation with partial derivatives:

0

        

t z Q s z P x z

; (7)

0 1 1 1 _         t y Q s y P x y

; y1



0,s,t



s; (8)

and 0 2 2 2 _         t y Q s y P x y

; y2



0,s,t



t. (9)

Now, it can be observed that:





 

   

 

   

 

   

 

   

 

   

 

    . , , , , , , , 2 , , , , 1 1 , , , , 1 2 , , , , 1 1 , , , , 1 2 2 , , , , 1 1 1 , , , , 1 2 1 1 2 1 2 1 2 1 2 1 2 1 2 1 t w Q s w P t y v v u f t y u v u f Q s y v v u f s y u v u f P t y Q s y P v v u f t y Q s y P u v u f x y y f w D t s x y v t s x y u t s x y v t s x y u t s x y v t s x y u t s x y v t s x y u t s x y v t s x y u t s x y v t s x y u x                                                                                         

The last relationship yields the equality:

0          t w Q s w P x w

; w



0,s,t



 f



y1



0,s,t

 

,y2 0,s,t



  

 f s,t . (10) Therefore, finally:



x st

 

wx st



z , ,  , , .

End of proof.

Other equalities can be proved analogously.

It is worth noting that the multiplicative operator defined by Eq. (6) can be considered as the exponential operator:

 

xDy

Mexp . ₍₁₁₎

Now, let us introduce the operator:

 



    0 0 ! : j j x j j D

M



. (12)

Note, that

 





n

n r r n r n x x n r x

M _  

     



  0

0 ; n0,1,2,.

Moreover,



0 1

 

0 1



0f x, ,s , ,sn  f x , ,s , ,sn

M









 .

3.4 The generalized operator of differentiation

We introduce the generalized multiplicative operator:











 

0 !

: j

j y j

D j x

G  . (13)

The following properties hold true:

(i) G



a₁f₁a₂f₂



a₁Gf₁a₂Gf₂; a₁,a₂R;

1 0, ,

, 2

1,f  s sn

f _{ } .

(ii) Gm xm; mZ0.

(iii) Gf₁



,s₀,,sn_1



f₁



x,Gs₀,,Gsn_1



.

(iv)



_



_



_



_

1 0

2

1 0

1 1 0

2

1 0

1

, , ,

, , , ,

, ,

, , ,

 



 _

n n n

n

Gs Gs x f

Gs Gs x f s s f

s s f G

  



 

.

(v) Let Msk gk



x,,s₀,,sn_1



. Then Gsk gk



x,,s0,,sn1



; k0,1,,n1.

We will prove the third equality (other equalities can be proved analogously).

Proof.

The second property of the multiplicative operator yields:





 





_

  

  













 

 

0

1 0

0 1

1 0

1

! , , ! , ,

, ,

j

n j y j j

j y j

n D s

j x s

D j x x

f s s

Mf     .

The replacement of the variable x by the expression x (what is possible) yields:





_

_

_

_





_{ }





_

_

    

 

 _{ }

  



_

_







 



 







0

1 0

0 1

0

1 0

1

! ,

, !

, ,

, , !

j

n j y j j

j y j j

n j

y j

s D j x s

D j x x

f s s f D j

x



_

_

_







 ,

what concludes the proof.

End of proof.

Definitions and properties of operators M, M0 and G (Eq. (11, 12, 13)) yield the following equality.

Corollary 1.

M M

G ₀ .

Thus, the generalized multiplicative operator G is a product of two noncommutative but multiplicative and exponential

operators. But the operator G is not an exponential operator. It is worth noting that exponential operators are widely used in

geometric-operator calculus [17, 21, 22]. We will demonstrate that the generalized multiplicative operator G can be effectively

4.

The expression of the solution to ODE in the operator form

Theorem 1.

The solution to the initial ODE problem Eq. (3, 4) can be expressed in the following form [19]:

yGs₀.

Without the loss of generality we will prove that the solution to the differential equation

   

  

dx dy y x P dx

y d

, ,

2 2 2

; yy



x,,s,t



; y



,,s,t



s;





t dx

t s x dy

x 



, , ,

(14)

reads:











D tD P



st



D



s j

x t

s x y

j

j t s

j







  



0

2 , ,

! ,

,

,  _  . (15)

Proof.

It is clear that D_yD tD_s P2



,s,t



D_t. The operator Dy satisfies all properties of the generalized operator of differentiation. Now, let





D s Ms

j x t s x z

j

j y j

 





0 !

, ,

, . Then,





Mt s MD Ms D Ms D dx

t s x dz

y y

x   



, , ,

.

Analogously,





_

_

_

_

_

_{ }



_

  

    

 

 

dx t s x dz t s x z x P Mt Ms M P t s MP t MD Mt D Mt D dx

t s x z d

y y

x

, , , , , , , , ,

, ,

, ,

, ,

2 2

2 2

2 _

  

 

.

Thus,





_

_{ }



_

  



 _ 

 

dx t s x

dz t s x

z x P dx

t s x

z

d , , ,

, , , , , ,

, ,

2 2

2 _{ }

  



.

Therefore,



x s t



Gs y



x s t



z ,, ,   ,, , and y



,,s,t



s;





t dx

t s x dy

x 



, , ,

.

End of proof.

Eq. (15) can be considered as the generalization of the Picard formula [23] describing the solution of an ordinary

differential equation in a power series form.





k k

n k

Gs dx

s s x y d



1 0, ,

,

, 

. (16)

Proof.

It is clear that Dk_ys0 s_k for k0,1,,n1. Then,

























. !

! !

! ,

, , ,

0

0

0 0

0

0 1

0

k j

k j y j

j

k j y j k

j

j y k j j

j y j k

x k

n k

Gs s D j x

s D j x s

D k j x s

D j x D dx

s s x y d

 



 

 

  











 







 



 

 



 



 

End of Proof.

Properties of multiplicative operators M and G yield the fact that the solution to the ODE initial value problem (Eq. (3, 4))

does satisfy the equality:



















, , ,

 

, , , ,



0.

, , , , ,

, , , ,

, , , ,

, , ,

1 1 0 1

0

2 1 0 1

0

1 0 1

1 0 1

0

 

  

 

    

   

   

 

  

  

 



n n n

n

n n n

n n

n

s s s x y s s P

s s s x y s s

s s x y s s s x y x

s s x y

 

 

 



  

  

 

  



For example, the solution to the initial value problem y x dx dy _{ }

; y



,,s



s takes the form



x s



 x



s

 

x



y , , 1 1exp . But the function y



x,,s

 

 s1

   

exp x  x1

 

:z x,,s



with the initial condition z



0,,s



s satisfies the following differential equation:







_{   }



0 , , ,

, ,

, _

        

s s x z s s x z x

s x

z _ 

  

.

4.1 Invariants of the generalized operator of differentiation and their properties

Definition 1. A function vv



x,,s₀,,sn₁



; vx,,s0,,sn1 is an invariant of Dy if

0



v

Dy . (17)

The set of invariants is denoted as



vD_yv0



:Ker D_y. Properties of invariants of D_y are listed below.

(i) Let v1,v2Ker D_y and

c

1

,

c

2



C. Then Dy

v v v v v c v

c ; ; Ker

2 1 2 1 2 2 1

1   . The proof follows from properties of Dy. (ii) Let  



z1,z2,,z_m



be a function of variables z1,z2,,zm and v1,v2,,vmKer Dy .

Then, 



v1,v2,,v_m



Ker D_y. Proof.



, ,





, ,



0

1 1

1   

 



y k

v z m

k k

m m

y D v

dz z z d v

v D

k k



 

End of proof.

(iii) Let vKer D_y and

1 0, ,

,

, 

 

n

s s x

f _{ } . Then, D_y

 

vf v

 

D_yf ; M

   

vf vMf ; G

   

vf vGf . Proof.

 

vf

 

D

v

f

v

D

f

v

 

D

f

D

_y



_y







_y



_y . Other equalities can be proved analogously.

End of proof. (iv) Let

1 0, ,

,

, 

 

n

s s x

f   . Then the operator





_

_

j y j

j

D j

G 



 



0 0

! :

~

0

 

 ; 0R (18)

yields the invariant G~ f Ker D_y

0 

 .

Proof.





_

_





_

_





_

_

_

_





_

_

_

_





























_

_





_

_

_

_

0 !

!

! !

1

! !

! !

~

0

1 0

0

1 0

0

1 0

1

1 0 0

1 0

0

0

0

0 0

0

 

 

 



 

 

    

  

   

     

  

    

    

 

 _



     

  

  

  















 



 







 



 





 





 



 





j

j y j j

j y j

j

j y j j

j y j

j

j y j

j y j y

j

j y j y

j

j y j y

y

f D j

f D j

f D j

f D j

f D j

f D j

D

f D j

D f

D j

D f G D



































End of proof.

Corollary 3. The replacement of a real complex number ₀ in the expression of

0

~



G by a symbol x yields the equality

G

G

~

_x



; (19)

moreover, GfKer D_y for

1 0, ,

,

, 

 

n

s s x

f _{ } .

Proof.





_

_









_

_





G D j x D

j x D

j x G

j

j y j j

j y j j

j y j

x 

       











 



 



0 ! 0 ! 0 !

~   

. Thus, G~_xf GfKer D_y for

1 0, ,

,

, 

 

n

s s x

f _{ } .

End of proof.

Note that the variable x is regarded as a constant in respect of the operator D_y.

Eq. (13), Eq. (18) and Eq. (19) yield the equality Dyy0; therefore

y

D

y



Ker

. (20)

Corollary 5. Let

1 0, ,

, , 2

1,f x s sn

f _{ } . Then Gf₁Gf₂ if and only if f₁ f₂. Proof.

If f₁f₂ then Gf₁Gf₂.

Now, let us assume that Gf₁Gf₂ . Then

















0 !

0

2 1 2

1  

  







j

j y j

f f D j x f

f

G  what requires that



f₁ f₂



0

D_yj holds for all j0,1,2,. But D_y0



f₁ f₂



 f₁ f₂ what concludes the proof.

End of proof.

Corollary 6. Let f



x,,s0,,s_n1



Ker D_y. The replacement of the symbol x by the symbol  produces a function



, ,s0, ,sn1



f    . Then, the following equality holds true:



, ,s0, ,sn1

 

 f x, ,s0, ,sn1



Gf     . (21)

Proof.

The function f can be expressed in the form











 

0 1 0

! ,

, , ,

j j j n

j x v s

s x

f   since f



x,,s0,,s_n1



Ker D_y, where

y

j D

v Ker . But then











 

0 1 0

! ,

, , ,

j j j n

j v s

s

f     what immediately yields Eq. (21).

End of proof.

Let y



x,,s,t



be the solution to differential equation (14) and y



x0,,s,t



:vx0



,s,t



:v ;





_

_

u t s u dx

t s x dy

x x x

 



: , , : , , ,

0 0

 

where x0R is not a singular point of differential equation (14). Then,

























C

a

u

v

a

D

kl

l k

l k kl y

0

,

Ker

Z

. (22)

It can be noted that Eq. (22) is not the only representation of Ker D_y in terms of u and v. In general, other representations (in other terms) do exist.

4.2 The canonical parameter of the generalized operator of differentiation

1

 

y

D . (23)

Corollary 7. The variable  is a canonical parameter of D_y because Dy1. Moreover, all other canonical parameters of D_y can be expressed in the form:

v  

 (24)

where vKer D_y.

Corollary 8. Let ₁ and ₂ be two canonical parameters of D_y. Then, y

D Ker

2 1 

 .

4.3 Structural expressions of the solution to ODE

We will consider several typical structural expressions of solutions. Let us assume that the function zz



,s₀,,sn_1



can be expressed in the form:



v vm



z, 1,, (25)

where

1 0, ,

, _

 

n

s s  

 ; vj vj



,s0,,sn1



Ker Dy; j1,2,,m. Then, properties of the generalized operator of differentiation yield:



x v vm



Gz , 1,, . (26)

Note that





D v v

j x v Gv

j

j y j

 

 





1 !



. Various particular cases of Eq. (25) could be considered. Several typical

examples are listed below.

Theorem 2.

Let us assume that s₀ can be expressed in the form:



v vm



s0, 1,, . (27)

Then,



xv vm



y , 1,, . (28)

Proof.

The proof follows from Eq. (13) and Eq. (26).

End of proof.

Eq. (28) also represents the structural expression of the solution in Eq. (13).

solve the initial problem defined by Eq. (3, 4) – the solution is automatically generated by Eq. (28). Since this result is of

fundamental importance, we denote the expression in Eq. (27) as the s₀ representation.

The natural question arises what is easier – to find invariants or to solve the initial problem using conventional techniques.

A discussion on these questions is provided in section 5.

Corollary 9. A particular case of the s₀ representation:

 







 

1 0

k

k k f

v

s  , (29)

where v_k v_k



,s0,,s_n1



Ker D_y for all k yields the solution to the ODE defined by Eq. (3, 4):

 







 

1 k

k k f x

v

y . (30)

Eq. (30) represents another structural expression of the solution in Eq. (13).

Note that







 

1 k

k k x

v

y if f_k

 

 k (what is a rather common situation).

Corollary 10.

Let us assume that the expression of the invariant vv



,s0



Ker D_y is given. Then Gvv



G,Gs0



what yields the

algebraic equation:







, , , 0, , 1



v x yx s sn

v   . (31)

Then the solution y could be expressed from Eq. (31). Thus it is possible (not always) to reduce the initial problem

defined by Eq. (3, 4) to the solution of the algebraic problem defined by equation Eq. (31).

It can be noted that other generalizations are also possible.

5.

Examples

A number of examples are given in this Section. We start from the most primitive examples and continue with more

demanding nonlinear ODE problems.

Example 1. Let us consider a differential equation 0 dx dy

; yy



x,,s



with the initial condition y



,,s



s. Then,

s

y D D

D  _ 0 ;











 

0 !

j

j j

D j x

G   . The invariant reads v

 

,s :s because Dyss. Then, the s-representation reads: s

s . Therefore, Gss; y



x,,s



Gs. Finally, y



x,,s



s; y



,,s



s and

























R

k Z k

k k

y

a

s

a

D

0

Example 2. Let us consider a differential equation 0

2 2

 dx

y d

; yy



x,,s,t



with initial conditions y



c;c,s,t



s;





t dx

t s x dy

x 



, , ,

. Then, Dy D tDs0Dt . The invariant reads: v



,s,t



st because



D tDs



st



tt0. Then the s-representation reads: s



s t



t . Now, Gs



s t



Gt (note that

  

G Gt xt t

G    ). Thus ystxt ; y



,,s,t



s ;





t dx

t s x dy

x 



, , ,

. Note that





t

dx t s x dy

Gt ,, ,  and



































0

;

,

Ker

0 0

,

0

0

x

x

a

t

t

x

t

s

a

D

kl

Z l k

l k kl

y



R

.

Example 3. Let us consider a differential equation P

 

x dx dy

1

 ; yy



x,,s



with the initial condition y



,,s



s. The generalized operator of differentiation reads: DyD P1

 

 Ds;





_

_{ }

_







 



0

1

! j

s j

D P D j x

G    . Now, let us define a

primitive function Pˆ1

 

x for P1

 

x :

 





 





 P zdz

Pˆ1 1 ; Pˆ1

 

 0 where



R and Pˆ1

 

 exists. Then, the invariant reads:

 

,s s Pˆ1

 



v   because



D P1

 

 Ds





sPˆ1

 





P1

 

 P1

 

 0 . Then the s-representation reads:

 



_{s P}ˆ₁



_Pˆ₁

 



s   . Thus, Gs



sPˆ1

 





Pˆ1

 

x . Finally y



x,,s







sPˆ1

 





Pˆ1

 

x ; y



,,s



s and

 

 





 































 1 1 0 0 1 0

ˆ

;

,

ˆ

ˆ

Ker

0

x

P

a

x

x

P

P

s

a

D

k

Z k

k k

y



R

.

Example 4. Let us consider a differential equation

 

 

y Q

x P dx dy

1 1

 ; yy



x,,s



with the initial condition y



,,s



s.

Then, _y

 

_{ }

D_s s Q P D D

1 1  

 ;







 

_{ }

 

 

 

 

 



0 1

1

! j

j s j

D s Q P D j x

G  _  . Let Pˆ1

 

x and Qˆ1

 

x be primitive functions for P1

 

x and

 

x

Q1 . Then, the invariant reads: v

 

,s Qˆ1

 

s Pˆ1

 

 , because

 

 



ˆ1

 

ˆ1

 



1

 

1

 

0

1

1 _{   }

   

 

    

 D Q s P P P

s Q P

D s . Now,

the s-representation can be expressed in the implicit form Qˆ1

 

s 



Qˆ1

 

s Pˆ1

 





Pˆ1

 

 . Therefore,

 



1

 

1

 





1

 



1 ˆ ˆ ˆ

ˆ _{s Q s P GP}

Q

G    ; Qˆ₁



G

 

s







Qˆ₁

 

s Pˆ₁

 





Pˆ₁



G

 





; Qˆ₁



y



x,,s





Qˆ₁

 

s Pˆ₁

 

 Pˆ₁

 

x . Finally



x s



Q



Q

 

s P

 

P

 

x



y ,,  ˆ₁1 ˆ₁  ˆ₁  ˆ₁ . The explicit solution exists if the inverse function Qˆ11

 

x

 _{does exist.}

Example 5. Let us consider a differential equation y2 dx dy_

; yy



x,,s



with the initial condition y



,,s



s. Then, s

y D s D

D  _  2 . It can be observed that

 

 

s s s v v

  

1

, because





_

_



_



_

0

1 1 1

1 2

2 2 2

2 _

        



   





s s s s s s s

s D s D v

Thus,

_{  }



_



_

s x xy

s x y G

Gs Gs s

s G s s

, , 1

, , 1

1

1 

 



      

 . The algebraic equation for the identification of y takes the form:

xy y s s

 

 1

1  . Finally







 s



x



s s

x y

1 ,

, . It can be noted that









_

_

0

1 ,

, 2 

  





 _

x s

s D

s D s x y

Dy s and



s



s y,,  and





_







































R

0 0

,

1

Ker

0

x

a

x

s

s

a

D

_k

Z k

k

k

y



.

Example 6. Let us consider a linear ordinary differential equation ₂ 0

2

 

 by

dx dy a dx

y d

; yy



x,,s,t



with initial conditions y



,,s,t



s ;





t

dx t s x dy

x 



, , ,

. Then, D_yD_ tD_s 



atbs



D_t and invariants becomev₁



,s,t

 

 t₂s

 

exp₁



; v₂



,s,t

 

 t₁s

 

exp ₂



where ₁ and ₂ are two different roots of the algebraic equation 2ab0. Note that Dyv1Dyv2 0. Then, the s-representation reads:



  



 

  

  

  

   

 

2 2

1 2

1 1

1 2

1 2

exp exp

exp

exp 

   

 

 t s t s

s .

Thus, the solution reads:





















 

    



 

   



 t s x s t x

y 2

2 1

1 1

2 1

2

exp exp

and



 





 



 





































 

R

kl l

k

l k

kl

y

a

t

s

t

s

a

D

0 ,

2 1

1

2

exp

exp

Ker













.

Example 7. Let us consider a linear ordinary differential equation ₂ 2 0 20 0

2

 

 y

dx dy dx

y

d _{ }

; yy



x,,s,t



with initial conditions y



,,s,t



s ;





t

dx t s x dy

x 



, , ,

. Then, Dy D tDs



t s



Dt 2 0 0

2 

   

 and invariants read











₀









₀



1 ,s,t  s t s exp

v ; v₂



,s,t

 

 t₀s

 

exp₀



; Dyv1Dyv2 0. The s-representation takes the form: s





s



t₀s





 

exp ₀



   

exp₀ 



t₀s

 

exp ₀



  

exp₀ .

Thus, the solution reads:

y





s





t





0

s



x









exp





0



x









and















 



 





    

  

 

   

 







R a s

t s

t s a

D _kl

l k

l k

kl y

0 ,

0 0

0

0 exp exp

Ker       

.

So far, the solution of trivial differential equations has been demonstrated in Examples 1 – 7. These examples were used