How to find general solution of PDE

Question

How to find general solution of equation $$ U_{xy}- \frac{U_x}{y} =0 ? $$

My approach: $$ U_{xy} = \frac{U_x}{y}. $$ Integrate w.r.t $x$ $$ y \ U_y = U + c $$ integrate w.r.t y

I don't know how to proceed further.

Answer 1

As noted by Chinny84, the integration constant $c$ should be a function of $y$. Then you get the equation $$ U_y=\frac{1}{y}\,U+C(y), $$ where $C(y)=c(y)/y$. This is a first order linear equation. Solving it and renaming the "constants" of integration, the general solution of the PDE is $$ U=\psi(x)\,y+\phi(y). $$