General solution of $u_{tt}+2au_{tx}+a^2u_{xx}=0$

Question

I am trying to solve the following partial differential equation (PDE) $$u_{tt}+2au_{tx}+a^2u_{xx}=0,$$

in which $a$ is a constant. By plugging this into Maple I have found the solution $u(x,t)=g(at-x)+xf(at-x)$. I see that it is some kind of d'Alembert Ansatz, but how could I solve this PDE from scratch?

Answer 1

Solve

$$\left(\partial_t+a\partial_x\right)^2u=0$$

$$\left(\partial_t+a\partial_x\right)\left(\partial_t+a\partial_x\right)u=0$$

$$\left(\partial_t+a\partial_x\right)v=0\implies v(t,x)=f(at-x)$$

$$v=\left(\partial_t+a\partial_x\right)u=f(at-x)$$