• Jan 16th 2009, 11:27 AM
mbbx5va2
Hi, i need to know how to solve this ODE. Please see the attachment. The solution is 2logx but i don't know to get it. Any help would be much appreciated. thanks
• Jan 16th 2009, 11:35 AM
Mush
Quote:

Originally Posted by mbbx5va2
Hi, i need to know how to solve this ODE. Please see the attachment. The solution is 2logx but i don't know to get it. Any help would be much appreciated. thanks

Let $\displaystyle u = \frac{dF}{dx}$. This should reduce it to a 1st order ODE of the form:

$\displaystyle \frac{1}{2}x^2 \frac{du}{dx} +xu =1$

Which can be written:

$\displaystyle \frac{du}{dx} +\frac{2}{x}u =\frac{2}{x^2}$

You can solve this for $\displaystyle u$ using an integrating factor! Giving a solution of the form:

$\displaystyle u = f(x) + C$

Then remember that:

$\displaystyle F = \int u dx$

Also, can I ask why you used $\displaystyle \partial$ instead of $\displaystyle d$ if it's an ODE? $\displaystyle \partial$ is reserved for PDEs, or Partial Differential Equations... ie... Differential Equations of functions of more than 1 variable.
• Jan 16th 2009, 01:42 PM
mbbx5va2
Sorry yes your right about the notation - i made a mistake. Thanks for the help i've managed to solve it.