Coupled partial differential equations I'm having trouble solving these coupled partial differential equations: 

$$\frac{\partial}{\partial t}f(x,t)-c\frac{\partial}{\partial x}f(x,t)-Ap(x,t)=0,$$ 
  $$\frac{\partial}{\partial t}p(x,t)+c\frac{\partial}{\partial x}p(x,t)+Af(x,t)=0,$$
  with $A,c$ real constants. 

What is the "trick" to solve these? I haven't tried a lot since I wouldn't know where to start looking. 
 A: $$\begin{cases}
\frac{\partial}{\partial t}f(x,t)-c\frac{\partial}{\partial x}f(x,t)-Ap(x,t)=0 \\ 
\frac{\partial}{\partial t}p(x,t)+c\frac{\partial}{\partial x}p(x,t)+Af(x,t)=0
\end{cases}$$
Regularised form with $\begin{cases} T=At \\ X=\frac{A}{c}x \end{cases} \quad\to\quad
\begin{cases}
\frac{\partial}{\partial T}f(X,T)-\frac{\partial}{\partial X}f(X,T)-p(X,T)=0 \\ 
\frac{\partial}{\partial T}p(X,T)+\frac{\partial}{\partial X}p(X,T)+f(X,T)=0
\end{cases}$
$$\begin{cases}
f_T-f_X-p=0 \\ 
p_T+p_X+f=0
\end{cases}$$
$p=f_T-f_X \quad\to\quad (f_{TT}-f_{XT})+(f_{XT}-f_{XX})+f=0$
$$\frac{\partial^2 f}{\partial T^2}-\frac{\partial^2f}{\partial X^2}+f(X,T)=0$$
Solving this hyperbolic PDE leads to $f(X,T)=f\left(At\:,\:\frac{A}{c}x \right)$
Then $\quad p(X,T)=\frac{\partial f}{\partial T}-\frac{\partial f}{\partial X}=p\left(At\:,\:\frac{A}{c}x \right)$
For example of solving see : Finding the general solution of a second order PDE
This method leads to the integral form of solution :
$$f(X,T)=\int c(s)e^{\sqrt{\alpha(s)-\frac{1}{2}}\:X +\sqrt{\alpha(s)+\frac{1}{2}} \:T  } ds$$
$c(s)$ and $\alpha(s)$ are arbitrary real or complex functions.
A: Hint.
We have
$$
\mathcal{D}_1 f = A p\\
\mathcal{D}_2 p = -A f
$$
then
$$
\mathcal{D}_2\mathcal{D}_1 f = A\mathcal{D}_2 p = -A^2 f\\
\mathcal{D}_1\mathcal{D}_2 p = -A \mathcal{D}_1f = -A^2p
$$
