General solution to the following PDE

Question

What is the general solution to the following PDE;

$y = y(x,t)$

$y + \frac{\partial y}{\partial x} + \frac{\partial^2 y}{\partial t^2} = 0$

Is it always the trivial solution? If so, how can I prove it?

Answer 1

Take $y(x,t) = \exp(-x) u(x,t)$ and the PDE becomes $\dfrac{\partial u}{\partial x} + \dfrac{\partial^2 u}{\partial t^2} = 0$ which is essentially the heat equation (with the usual roles of $x$ and $t$ reversed).