# Differential Equation and Initial Value Problem with solution u(x)

It's my first week dealing with Differential Equations, and I am totally lost regarding the following question. Any help would be very much appreciated!

u(x)is a solution to initial value problem:

$$xy'=y-xe^{\frac{y}{x}}$$

y(e)=0

a. $$u(e^e)=e^e$$

b. $$u(e^e)=2^e$$

c. $$u(e^e)=-e^e$$

d. $$u(e^e)=e^2$$

e. $$u(e^e)=e^{-e}$$

• Are you supposed to guess the correct result with just a glance or do you compute the solution first and then check the result? In the latter case one could also just have asked for the value of $u(e^e)$. – LutzL Jul 11 at 7:30

You should recognize that the main intermediate expression of your equation is $$v=y/x$$. Insert that to get everything to contract nicely to $$v'=\frac{xy'-y}{x^2}=-\frac{e^v}x,$$ which can now be solved as separable ODE.