Mathematical Physics Vol 1

Chapter 7. Partial differential equations

344

Thus, the condition is necessary. The condition is sufficient . If the function ψ satisfies the equation (7.21), then it also satisfies the equation (7.20), which means that it represents a first integral of the initial system (7.16), under the condition that the functions y i , i = 1 ,..., n , represent a set of solutions of the given system.

R Note that if the functions ψ i ( i = 1 ,..., m ) are m first integrals of the initial system, then any differentiable function of the form

F ( ψ 1 ,..., ψ m ) .

is also a first integral of this system.

7.3.3 Symmetrical form of a system of ordinary differential equations Observe a system of ordinary differential equations (ODE) (7.16)

d y i d x

= f i ( x , y 1 ,..., y n ) , i = 1 , 2 ,..., n .

(7.22)

Solving this system for d x , we can write in the form

d x 1

d y 1 f 1

d y n f n

(7.23)

= ··· =

=

.

If we now introduce the following substitutions

x = x 1 , y 1 = x 2 ,..., y n = x n + 1 ,

and then divide the relations (7.23) by some function X 1 , which is not identical to zero in the observed region D , we obtain d x 1 X 1 = d x 2 X 2 = ··· = d x n + 1 X n + 1 (7.24) the so called symmetrical form of the system of ordinary differential equations (7.22). Here we have introduced the substitutions

X i + 1 = f i X 1 , i = 1 , 2 ,..., n .

The points M 0 ( x 0

0 n + 1 ) at which X i ( M 0 )= 0 are called singular points of the system

1 ,..., x

(7.24).

R Note that the unambiguity of the solution is violated at these points, so they should not be taken as initial conditions.