Integration within a DiffEQ problem

[Adrynalyne]

Homework Statement

Solve the given differential equation by using an appropriate substitution.

Homework Equations

$$(x^{2}+xy+3y^{2})dx-(x^{2}+2xy)dy=0$$
$$y=ux, dy=udx+xdu$$

The Attempt at a Solution

$$(x^{2}dx+ux^{2}dx+3u^{2}x^{2}dx)-(ux^{2}dx+x^{3}du+2u^{2}x^{2}dx+2ux^{3}du)=0$$
After combining, cancelling and moving terms into their appropriate places, I get:
$$\frac{dx}{x}=\frac{2u+1}{u^{2}+1}du$$


This is where I get stuck, I am unable to integrate the right hand side. Can anyone help me out a little?

Thanks.

 

[Dick]

Split it into two parts. 2u/(u^2+1) looks easy by a substitution and 1/(u^2+1) looks like an arctan.

 

[Adrynalyne]

I cannot believe I didn't see that.

Thanks!