Mathematical Physics Vol 1

7.3 Linear and quasilinear first order PDE

343

7.3.2 A general method for integrating linear first order PDE. First integral

In this section we will show the relation between linear first order PDE and a system of ordinary differential equations of the form d y i d x = f i ( x , y 1 ,..., y n ) , i = 1 , 2 ,..., n , (7.16) where x is the independent variable, and y i = y i ( x ) are unknown functions of the variable x . Assume that the functions f i ( i = 1 ,..., n ), which depend on n + 1 independent variables x , y 1 ,..., y n , are differentiable in some closed region D , of the space R n + 1 . In the theory of ordinary differential equations, it has been proved (see, for example, [42]) that under the given assumptions, only one integral curve of the system (7.16) passes through a given point M 0 ( x 0 , y 0 1 ,..., y 0 n ) , of the region D , i.e. that there exists an unambiguously defined set of functions y i ( x ) that represents the solution of the initial system (7.16), and satisfies the conditions y i ( x 0 )= y 0 i , i = 1 ,..., n . The first integral of the system (7.16) is every function ψ ( x , y 1 ,..., y n ) that is not identically reduced to a constant when y i is replaced by a set of solutions of system (7.16). It is common to call the function of the form ψ ( x , y 1 ,..., y n )= c (7.17) the first integral, where c is an arbitrary constant. Theorem23 The necessary and sufficient condition for the function ψ ( x , y 1 ,..., y n )= c to be the first integral of the system (7.16) is that it satisfies the linear first order partial differential equation ∂ψ ∂ x + n ∑ i = 1 f i ( x , y 1 ,..., y n ) ∂ψ ∂ y i = 0 . (7.18)

Proof The condition is necessary . Assume that the function ψ ( x , y 1 ,..., y n ) is the first inte gral of the system (7.16). Then this function, for the set y i that represents the solutions of the initial system (7.16), is reduced to a constant, namely ψ ( x , y 1 ,..., y n )= c . (7.19)

As c is a constant, it follows that the total differential of this function is

n ∑ i = 1

∂ψ ∂ x

∂ψ ∂ y i

d y i d x

d ψ =

= 0 ,

(7.20)

+

that is, using the relations (7.16), ∂ψ ∂ x + n ∑ i = 1

∂ψ ∂ y i

f i ( x , y 1 ,..., y n )= 0 .

(7.21)