For reference, the entire equation to be solved for $u(x,y)$ is:
- $A= -2x^2-8xy-8y^2+42x-14y$
- $B= -5x^2-20xy-20y^2+105x-35y$
- $C= 3x^2+12xy+12y^2-63x+21y$
- $E= 28x+56y$
- $K= -14x-28y$
where $Au_{xx}+Bu_{xy}+Cu_{yy}+Eu_x+Ku_y = 0$
I already have done the majority of the work and reduced the problem to:
- $(r^2+7s)u_{rs} - (2r)u_s = 0$
where $r = (2y+x)$ and $s = (y-3x)$ (note $r$ and $s$ are functions of $x$ and $y$)
But I'm now unsure how to proceed with integrating from this point and arriving at $u(r,s)$, from where I can easily change into $u(x,y)$.
If $v = u_s$, the reduced problem is $$ (r^2 + 7 s) \dfrac{\partial v}{\partial r} - 2 r v = 0 $$
Solve this as an ordinary differential equation (treating $s$ as a constant parameter). The arbitrary constant becomes an arbitrary function of $s$. Then integrate with respect to $s$, with an arbitrary constant that is an arbitrary function of $r$.