# Thread: Integrating factor on Exact, Homogeneous DE

1. ## Integrating factor on Exact, Homogeneous DE

Problem: $\displaystyle y*(1+x)+2xy'=0$

I have rearranged the equation to put it in exact form, resulting in the equation:
$\displaystyle (y+xy)dx+2xdy=0$

After that I found my $\displaystyle \frac{\partial M}{\partial y}$ and $\displaystyle \frac{\partial N}{\partial x}$

This leads to a complicated integrating factor of $\displaystyle e^{-\frac{1}{2}(ln|x|-x)}$

Am I on the correct track?

2. ## Re: Integrating factor on Exact, Homogeneous DE

Originally Posted by cheme
Problem: $\displaystyle y*(1+x)+2xy'=0$

I have rearranged the equation to put it in exact form, resulting in the equation:
$\displaystyle (y+xy)dx+2xdy=0$

After that I found my $\displaystyle \frac{\partial M}{\partial y}$ and $\displaystyle \frac{\partial N}{\partial x}$

This leads to a complicated integrating factor of $\displaystyle e^{-\frac{1}{2}(ln|x|-x)}$

Am I on the correct track?
Dividing throughout by $\displaystyle 2x$, we get

$\displaystyle y'+\left(\frac{1+x}{2x}\right)y=0$

The integrating factor is:

$\displaystyle exp\left[\int \left(\frac{1+x}{2x}\right)dx\right]$

$\displaystyle =exp\left[\int \left(\frac{1}{2x}+\frac{1}{2}\right) dx\right]$

$\displaystyle =exp\left(\frac{1}{2}ln\ x+\frac{x}{2}\right)$

$\displaystyle =exp\left(\frac{1}{2}ln\ x\right)*exp\left(\frac{x}{2}\right)$

$\displaystyle =\sqrt{x}e^{x/2}$

(Note that this isn't an exact differential equation as $\displaystyle \frac{\partial M}{\partial y} \neq\frac{\partial N}{\partial x}$.)

3. ## Re: Integrating factor on Exact, Homogeneous DE

I did not see to divide through by 2x. If I do that, isn't it just a separable equation?

4. ## Re: Integrating factor on Exact, Homogeneous DE

Originally Posted by cheme
I did not see to divide through by 2x. If I do that, isn't it just a separable equation?
Yes, this is a separable equation.