PDE separation of variable problem

[JamesGoh]

Homework Statement

Using the method of separation of variables, obtain a solution of the following PDE subject to
the given conditions

$\frac{du}{dx}+y\frac{du}{dy}=(2x+y)u$

$u(x,1)=5e^{x^{2}-x}$

Homework Equations

The Attempt at a Solution

See my attached working

the answer is $u(x,y)=5ye^{y-1+x^{2}-x}$

Im not sure if I am getting anywhere near close in my working out. Have double checked the arithemetic and everything seems fine so far

 

[JamesGoh]

hi guys

an update on the problem. I managed to solve it.

The issue was that I didn't consider the arbirtrary integration constant, C, when doing the integral of the Y term.