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

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- Jan 16th 2009, 11:27 AMmbbx5va2Please help! ODE question
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 AMMush
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 PMmbbx5va2
Sorry yes your right about the notation - i made a mistake. Thanks for the help i've managed to solve it.