Multivariate coupled partial differential equation

Question

Are there any general methods to solve linear coupled multivariate partial differential equations of the first order?

For an example:

Are these coupled system of differential equations of two variables solvable?

\begin{align} a \psi(x,y)+ (b x-g) \partial_{x}\phi(x,y)+(cy-g) \partial_{y}\phi(x,y)-\left(m+gx\right)\phi(x,y)-\left(m+gy\right)\phi(x,y)&=0\\ a \phi(x,y)+(bx+g) \partial_{x}\psi(x,y)+(cy+g) \partial_{y}\psi(x,y)-\left(m-gx\right)\psi(x,y)-\left(m-gy\right)\psi(x,y)&=0 \end{align}

Here $a,b,c,g$ and $m$ are constants.

Any hint toward a solution and also any references for these kinds of equations would be really helpful.

Answer 1

Methods for explicitly solving simultaneous first order PDE can be found in:

A.R. Forsyth Theory of Differential Equations (Cambridge, 1906). (Specifically, Part IV Partial Differential Equations, Volume V, chapter XI.)

The book is widely available for download, free of charge.

It could be worth trying the method described by

M. Kourensky, "A Method of Integrating the General form of a System of Partial Differential Equations of the First Order in two Dependent and two Independent Variables" Proc. London Math. Soc. (1930) s2-31 pp. 407-416

Answer 2

The answer to your first question is a clear no. There are only a few tiny classes of PDEs which have any chance at being solvable (in the sense of being able to write down a solution explicitly).

Luckily, your example is a linear first order equation, which are such a class. Essentially it is possible to find curved coordinates along which your PDE reduces to a family of ODEs. Look up the method of characteristics for details.