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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)+(by+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.

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.

Are there any general methods to solve coupled multivariate partial differential equations?

For an example:

Is this coupled system of differential equations of two variable 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)+(by+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.

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)+(by+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.

Are there any general methods to solve coupled multivariate partial differential equations?

For an example:

Is this coupled system of differential equations 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)+(by+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 helpul.

Are there any general methods to solve coupled multivariate partial differential equations?

For an example:

Is this coupled system of differential equations of two variable 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)+(by+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.