Frobenius method solution for linear ODE

[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.

Thank you in advance for your time and effort



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.

 

[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" .

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

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/" .

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:
$$x\int e^xx^{-2} dx$$