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