Q. show that u(x, y)=xf(2x+y) is a general solution of x∂u/∂x-2x ∂u/∂y=u

Question

This question is about partial differential equation.

Answer 1

Q. show that u(x, y)=xf(2x+y) is a general solution of x∂u/∂x-2x ∂u/∂y=u

We have u(x,y) = x.f(2x+y) = x.f(v), where v = 2x + y

∂u/∂x = f(2x+y) + x(∂f/∂x) = f(2x+y) + x(∂f/∂v)(∂v/∂x)= f(2x+y) + x.fv.2  (fv = ∂f/∂v)

x.∂u/∂x = x.f(2x+y) + 2x^2.fv = u + 2x^2.fv

∂u/∂y = x(∂f/∂y) = x(∂f/∂v)(∂v/∂y) = x.fv.1  (fv = ∂f/∂v)

2x.∂u/∂y = 2x^2.fv

x.∂u/∂x​ - 2x.∂u/∂y​ = u + 2x^2.fv​ - 2x^2.fv

x.∂u/∂x​ - 2x.∂u/∂y​ = u