# Separation of Variables: How to integrate (x+2y)y'=1 y(0)=2?

## Homework Statement

Use separation of variables to solve (x+2y)y'=1 y(0)=2

## Homework Equations

u=2y+x >>I did not know how to start this, so i looked at the back of the book and it said to use that substitution
y=(u-x)/2, du=2dy+dx, dy=(du-dx)/2

## The Attempt at a Solution

so i got the following:

dy/dx=1/(x+2y)
(du-dx)/(2dx)=1/(x+2(u+x)/2)
(du-dx)/(2dx)=1/u

I could not separate the variables from here. Also, according to the back of the book, the answer is supposed to be 2y-2ln|2+x+2y|+4+2ln2=0. But the term -2ln|2+x+2y| has both x and y variables, so aren't the variables not separated? That still qualifies as a solution by Separation of Variables?

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Dick
Homework Helper
dx/dx=1, so you last equation is du/dx-1/2=1/u. Can you separate u and x in that? And, no, the equation doesn't separate in y and x, but it does in u and x and I think that counts as a 'separation of variables' solution after the substitution.

I finally solved it, thank you!!

I got a different answer using linear differential equations.

dy/dx = 1/(x + 2y)
dx/dy = x + 2y

dx/dy - x = 2y

The answer i got was: x = -2y -2 + 6e^(y-2)

Differentiating it again returns me to the original differential