Consider this set of coupled equations:
$\frac{\partial}{\partial x}f(x,y)=a(x)f(x,y)+b(x)g(x,y)$;
$\frac{\partial}{\partial x}g(x,y)=c(x)f(x,y)+d(x)g(x,y)$,
with boundary condition $f(x,x)=h(x)$ and $g(x,x)=k(x)$, i.e. the values of $f$ and $g$ are known when the two arguments are equal. Here $a(x),b(x),c(x),d(x),h(x),k(x)$ are known functions.
Is there any known analytical solution to this set of equation?
If it would help, if we define $F(x)=\int_0^x f(x,y)\exp[-\gamma_f(x-y)]dy$, and $G(x)=\int_0^x g(x,y)\exp[-\gamma_g(x-y)]dy$, then $F(x)$ and $G(x)$ are also known.