Question

(D^2 - α^2D'^2)z =x

Answer 1

A partial solution of an equation $u_{tt}(x,t) – a^2u_{xx}(x,t)=f(x,t)$ with the initial conditions $u(x,t)=0$ is given as follows:

$u(x,t)=\frac{1}{2a}\int\limits_{0}^{t}\int\limits_{x-a(t-\tau)}^{x+a(t-\tau)}f(\xi,\tau) d\xi d\tau$

Replacing $f(x,t)$ with $x$, we have

$u(x,t)=\frac{1}{2a}\int\limits_{0}^{t}\int\limits_{x-a(t-\tau)}^{x+a(t-\tau)}\xi d\xi d\tau = \frac{1}{2a}\int\limits_{0}^{t} (\frac{(x+a(t-\tau))^2}{2}-\frac{(x-a(t-\tau))^2}{2}) d\tau = \int\limits_{0}^{t} x(t-\tau)d\tau=xt^2/2$ The general solution of a linear non-homogeneous equation is a sum of a partial solution and a general solution of the homogeneous equation.

The final answer will be

$u(x,t)=xt^2/2 + g(x-at)+h(x+at)$