How we can solve linear system of equations

Question

How we can solve following linear system of equations?

$A_{1x}=ia(A_1+A_2)$,

$A_{2x}=ia(A_1-A_2),$

$A_{1t}=(ia/2)A_1-A_2$,

$A_{2t}=A_1+(ia/2)A_2$,

where $A_1=A_1(x,t)$ and $A_2=A_2(x,t)$ and $a$ is a parameter independent of $x$ and $t$.

And $A_{1x}$ means partial derivative of $A_{1}$ w.r.t $x$.

Answer 1

Hint: add the first two equations together, then take the derivative with respect to $x$. You get $$A_{1x}+A_{2x}=2iaA_1\\(A_1+A_2)_{xx}=2iA_{1x}=-2a^2(A_1+A_2)$$ Now do the subtraction: $$A_{1x}-A_{2x}=2iaA_2\\(A_1-A_2)_{xx}=2iA_{2x}=-2a^2(A_1-A_2)$$ Let's call $f=A_1+A_2$ and $g=A_1-A_2$. Solve for $f$ and $g$ then get $A_1=\frac{f+g}{2}$ and $A_2=\frac{f-g}{2}$. This will give the $x$ dependent part. Proceed similarly with the last two equations to get the time dependence.