1st order ODE, Reduction to Separable Form Help required

• Feb 27th 2013, 08:35 PM
Nuingaer
1st order ODE, Reduction to Separable Form Help required
Hey guys,
I'm very new to this form of solving differential equations and I've come to a stump in a problem.

The problem is simply to find the general solution to $\displaystyle y'=(4x^2+y^2)/(xy)$

I have a guide that also shows the steps to the solution, but the area I am stuck on is identical to the guide, where i don't see how it got from one step to the next.

This is what I have so far;

Set$\displaystyle u=\frac{y}{x}, y=xu, y'=u+xu'$

$\displaystyle y'=\frac{4x^2}{xy}+\frac{y^2}{xy}$

$\displaystyle y'=\frac{4x}{y}+\frac{y}{x}$

$\displaystyle u+xu'=\frac{4}{u}+u$

Therefore;

$\displaystyle xu'=\frac{4}{u}$

Now, at this point I assumed it would be very easy to separate the variables to;

$\displaystyle uu'=\frac{4}{x}$

And solve by integration. However, the guide goes from

$\displaystyle xu'=\frac{4}{u}$

to

$\displaystyle 2u du = \frac{8}{x} dx$
My question is... what step am i missing that u becomes 2u and 4/x becomes 8/x?

• Feb 27th 2013, 10:12 PM
Prove It
Re: 1st order ODE, Reduction to Separable Form Help required
Quote:

Originally Posted by Nuingaer
Hey guys,
I'm very new to this form of solving differential equations and I've come to a stump in a problem.

The problem is simply to find the general solution to $\displaystyle y'=(4x^2+y^2)/(xy)$

I have a guide that also shows the steps to the solution, but the area I am stuck on is identical to the guide, where i don't see how it got from one step to the next.

This is what I have so far;

Set$\displaystyle u=\frac{y}{x}, y=xu, y'=u+xu'$

$\displaystyle y'=\frac{4x^2}{xy}+\frac{y^2}{xy}$

$\displaystyle y'=\frac{4x}{y}+\frac{y}{x}$

$\displaystyle u+xu'=\frac{4}{u}+u$

Therefore;

$\displaystyle xu'=\frac{4}{u}$

Now, at this point I assumed it would be very easy to separate the variables to;

$\displaystyle uu'=\frac{4}{x}$

And solve by integration. However, the guide goes from

$\displaystyle xu'=\frac{4}{u}$

to

$\displaystyle 2u du = \frac{8}{x} dx$
My question is... what step am i missing that u becomes 2u and 4/x becomes 8/x?