Question

solve px+qy=y

Answer 1

$px+qy=y$

$\frac{\partial z}{\partial x}x+\frac{\partial z}{\partial y}y=y$

The auxiliary equations is:

$\frac{dx}{x}=\frac{dy}{y}=\frac{dz}{y}$

A first characteristic equation comes from

$\frac{dx}{x}=\frac{dy}{y}$

$\ln{|x|}=\ln{|y|}+\ln C_1$

$x=C_1y$

A second characteristic equation comes from

$\frac{dy}{y}=\frac{dz}{y}$

$dy=dz$

$y=z+C_2$

General solution of the PDE on the form of implicit equation:

$\Phi(C_1,C_2)=0$.

Answer: $\Phi(C_1,C_2)=0$ , where $C_1=\frac xy$, $C_2=y-z$.