Mathematical Physics Vol 1

Chapter 7. Partial differential equations

346

7.3.5 General solution of linear non-homogeneous first order PDE

The solution of the non-homogeneous first order partial differential equation can be obtained in a similar way as for the homogeneous case. Observe the non-homogeneous first order partial differential equation n ∑ i = 1 a i ∂ u ∂ x i = b , (7.29) where u is an unknown continuous function of variables x i , i = 1 ,..., n . The values a i and b are assumed to be functions of the form a i = a i ( x 1 ,..., x n ) and b = b ( x 1 ,..., x n , u ) . Let us search for a general solution in the form v ( x 1 ,..., x n , u )= 0 . (7.30) By differentiating, we obtain

∂ v ∂ x i ∂ v ∂ u

∂ v ∂ x i

∂ v ∂ u

∂ u ∂ x i

∂ u ∂ x i

= 0 ⇒

(7.31)

= −

+

.

∂ u ∂ x i

Substituting

, from (7.31) into (7.29), we obtain a 1   − ∂ v ∂ x 1 ∂ v ∂ u   + ··· + a n   − ∂ v ∂ x n ∂ v ∂ u

  = b ,

∂ v ∂ u

that is, multiplying by −

,

∂ v ∂ x 1

∂ v ∂ x n

∂ v ∂ u

a 1

+ ··· + a n

= − b

,

or

∂ v ∂ x 1

∂ v ∂ x n

∂ v ∂ u

a 1 (7.32) This expression represents a homogeneous first order differential equation. Thus, any solution of equation (7.32), which contains the variable u , when equated to 0, gives the solution of equation (7.29) in the form (7.30). To this equation corresponds the system of ordinary equations d x 1 a 1 = ··· = d x n a n = d u b . (7.33) First integrals are of the from ψ 0 ( x 1 ,..., x n , u )= c 0 , .. . = .. . ψ n − 1 ( x 1 ,..., x n , u )= c n − 1 . (7.34) The general integral is of the form F ( ψ 0 ,..., ψ n − 1 )= 0 , (7.35) where F is a differentiable function of its arguments. + ··· + a n + b = 0 .