Mathematical Physics Vol 1

7.3 Linear and quasilinear first order PDE

345

7.3.4 General solution of the linear homogeneous first order PDE

Observe the linear homogeneous first order partial differential equation of the form

n ∑ i = 1

∂ u ∂ x i

L [ u ] ≡

a i

= 0 .

(7.25)

Coefficients a i = a i ( x 1 ,..., x n ) are assumed to be differentiable functions in a region D of the n –dimensional space, and not annulled simultaneously at point M 0 ( x 0 1 ,..., x 0 n ) . As we have previously shown, we can assign to this partial equation a system of ordinary differential equations, in symmetric form d x 1 a 1 = d x 2 a 2 = ··· = d x n a n . (7.26) According to the previous theorem, the first integrals ψ i = c i of the system (7.26) are solutions of the partial equation (7.25). Thus, solving the initial partial equation (7.25) comes down to solving the system (7.26), i.e. finding its first integrals. From the system (7.26) we can determine n − 1 mutually independent first integrals ψ i ( x 1 ,..., x n )= c i , i = 1 , 2 ,..., n − 1 , (7.27) where c i are arbitrary constants. We have already noted that if ψ i ( i = 1 ,..., ( n − 1 )) are first integrals, and thus each differ entiable function of the following form is also a first integral F ( ψ 1 ,..., ψ n − 1 ) . Thus, F represents the general solution of the partial differential equation (7.25). This can be easily proved. Assume that the general solution of equation (7.25) has the form u = F ( ψ 1 ,..., ψ n − 1 ) , where F is an arbitrary function, and ψ i = c i ( i = 1 ,..., n − 1) are first integrals of the system (7.26). It follows that L [ F ]= a 1 ∂ F ∂ψ 1 ∂ψ 1 ∂ x 1 + ··· + ∂ F ∂ψ n − 1 ∂ψ n − 1 ∂ x 1 + + a 2 ∂ F ∂ψ 2 ∂ψ 2 ∂ x 2 + ··· + ∂ F ∂ψ n − 1 ∂ψ n − 1 ∂ x 2 + ··· + + a n ∂ F ∂ψ n ∂ψ n ∂ x n =

   a 1 ∂ψ 1 ∂ x n | {z } L [ ψ 1 ]  ∂ψ 1 ∂ x 1 + ··· + a n

(7.28)

 

∂ F ∂ψ 1

+ ··· +

=

   X 1 |

  

∂ F

∂ψ n

∂ψ n

− 1

− 1

+ ··· + a n

= 0 ,

+

∂ψ n

∂ x 1

∂ x n

− 1

{z

}

L [ ψ n

− 1 ]

which was to be proved.