How do I solve the differential equation of the type $$ x u_x + y u_y = f(x, y) $$ For example, let $f(x, y) = xy$. Using following method, I found $F(x,y) = F(x/y)$ is solution to homogeneous equation
What method can I use to find the particular solution of the given DE? When $f(x,y) = xy$, I put $$ u_y = cx \implies u(x, y) = cxy $$ Putting this on DE, I found $c=1/2$. Is there general way to solve for particular solution of PDE?