find general solution of PDE: $ xu_{xy}+u_y=2xe^y $

Question

$$ xu_{xy}+u_y=2xe^y $$

I solved this equation like following:

Divided the equation by $x$: $$ u_{xy}+ \frac {1}{x}u_y = 2e^y $$

Then integrated with respect to $x$: $$ u_y + \ln(x)u_y = 2xe^y+f(y) $$

Then: $$ u_y (1+\ln(x)) = 2xe^y +f(y)$$

Then integrated with respect to $y$: $$ u = \frac{2x}{1+\ln(x)} e^y + \int f(y) + h(y)$$

However I am thinking that my 3rd step is incorrect. Can you help guys, if I am doing something wrong?

Answer 1

In fact the second step is incorrect: Since $u$ may depend on $x$, integrating $f_x u_y$ in general does not give $f u_y$.

Hint We can proceed as follows: Since $u$ only appears in the equation differentiated by $y$, we can produce a lower-order equation by writing $v := u_y$: $$x v_x + v = 2 x e^y .$$

Additional hint The left-hand side is just $(x v)_x$, so we can immediately integrate to obtain $x v$, thus $v$, and then integrate again to obtain $u$.