Answer to Question #217123 in Differential Equations for birch

Question

d^2u/dx^2 - d^2u/dy^2 = 2(e^(-x)) + 3sin2y

Answer 1

$\frac{d^2u}{dx^2} - \frac{d^2u}{dy^2} = 2e^{-x} + 3\sin 2y$

This is an inhomogeneous linear PDE. Its general solution can be obtained as a sum of a partial solution and the general solution of the corresponding homogeneous linear PDE.

Consider the homogeneous linear PDE:

$\frac{d^2u}{dx^2} - \frac{d^2u}{dy^2} = 0$

This is a well-known wave equation, it has the general solution $u_{hom}= f_1(x+y) + f_2(x-y)$ , where $f_1$ and $f_2$ are arbitrary functions.

Let's find a partial solution of the form $u_p=g(x)+h(y)$ , where $g$ , $h$ are unknown functions.

$\frac{d^2u_p}{dx^2} - \frac{d^2u_p}{dy^2} =g''(x)-h''(y)=2e^{-x} + 3\sin 2y$

Hence

$g''(x)=2e^{-x}$ and $g(x)=2e^{-x}$ (partial solution)

$-h''(y)=3\sin 2y$ and $h(y)=\frac{3}{4}\sin 2y$ (partial solution)

Therefore, the general solution to the given equation is

$u=u_p+u_{hom}=2e^{-x}+\frac{3}{4}\sin 2y+f_1(x+y)+f_2(x-y)$