Question

Q15. Form a partial differential equation for z =
z f(?)
(A) px - qy = 0
(B) px + qy = 0
(C) py - 4x = 0
(D) py + qx = 0​

Answer 1

Answer:

D=∂∂x and D′=∂∂y.

This PDE can be rewritten in factored form as

(D−2D′+1)(D+D′+5)z=0.

Since a general solution to (aD+bD′+c)z=0 (with a,b,c constants) can be written as z=e−cx/aϕ(ay−bx) for some one-variable arbitrary function ϕ when a≠0 (or z=e−cy/bϕ(ay−bx) if b≠0), we conclude that a general solution to the PDE in question is

z=e−xf(y+2x)+e−5xg(y−x)

for some arbitrary one-variable functions f and g.