Convert second order PDE $u_{tt} = u_{xx} + u$ to a system of first order PDE's I am attempting to convert 
$$ u_{tt} = u_{xx} + u$$
to a system of first order PDE's.  I believe that the system will require 3 equations, one for each of $u, u_t,$ and $u_x.$  Here is my attempt:
\begin{equation} \frac{\partial}{\partial t} \begin{pmatrix} u \\ u_t \\ u_x \end{pmatrix} -  \frac{\partial}{\partial x} \begin{pmatrix} u \\ u_x \\ u_t \end{pmatrix} = \begin{pmatrix} u_t - u_x \\ u \\ 0 \end{pmatrix} \end{equation}
This doesn't feel correct.  If anyone is familiar with a standard way to do this, any help would be appreciated.  

This is a wave equation and you need to change variables as follows: p = x+t, q = x-t. You will obtain:
$$ -4\frac{d^2u}{dpdq}=u(p,q)$$
And then set $$ v= \frac{du}{dq} $$ and another equation $$\frac{dv}{dp} = -u(p, q)/4 $$
