# First Integral

## First Integral

The first integral of a system of ordinary differential equations

is a relation of the form

*ϕ* (*x, y*_{1}, …, *y _{n}*) =

*C*

where *C* is an arbitrary constant. The left side of the relation remains constant when any solution *y*_{1} = *y*_{1}(*x*), …, *y _{n}* =

*y*) of the system is substituted but is not a fixed constant. Geometrically, the first integral is a family of hypersurfaces in the (

_{n}(x*n*+ 1)-dimensional space

*Oxy*

_{1}…

*y*, in each of which there exists a subfamily of the integral curves of the system. For example,

_{n}*y*

^{2}+

*z*

^{2}=

*C*

^{2}(circular cylinder) is a first integral of the system

*dy/dx*=

*z*,

*dz/dx*= –

*y;*the integral curves

*y*=

*C*sin (

*x*–

*x*

_{0}) and

*z*=

*C*cos (

*x*–

*x*

_{0}) are helices on the cylinders (see Figure 1). If

*k*independent first integrals

*ϕ*(

_{i}*x, y*

_{1}…,

*y*) =

_{n}*C*(

_{i}*i*= 1, …,

*k*;

*k*<

*n*) of a system are

known, the system’s order can in general be decreased by *k* units. If *k* = *n*, the general integral of the system is obtained without integration.

### REFERENCE

Stepanov, V. V.*Kurs differentsial’nykh uravnenii*, 8th ed. Moscow, 1959.