# Frobenius method solution for linear ODE

1. Jun 17, 2010

### zokomoko

I've been given the ODE:
x^2 u''-x (x u'-u)=0
Solve.

It's suppose to be an example in which a logarithmic term is required for the general solution.
I would be glad if someone could look at what I've done and see if my solution is correct / incorrect.

p.s: I've attached a zip file which contains a WMF type file of the problem, if anyone has mathtype it's easier to read than the attached GIF image.

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2. Jun 26, 2010

### ross_tang

Your 1st solution of y = x is correct, but the other solution is wrong.

Please refer to the below question in http://www.voofie.com" [Broken].

http://www.voofie.com/content/86/how-to-solve-this-2nd-order-linear-differential-equation/" [Broken]

The question solves an inhomogeneous version of your question using http://www.voofie.com/content/84/solving-linear-non-homogeneous-ordinary-differential-equation-with-variable-coefficients-with-operat/" [Broken].

If there is an inhomogeneous function of f(x), the solution should be:
$$\Rightarrow y=C_1x\int e^xx^{-2} dx+ C_2 x+x\int \left(e^xx^{-2} \int e^{-x} x f(x)dx\right) \, dx$$

And you can see, the fundamental solution that you are missing is:
$$x\int e^xx^{-2} dx$$

Last edited by a moderator: May 4, 2017