Solving the Bessel Equation with Initial Conditions and Bessel Functions

[makasx]

this is my final exam question, I can't figure how to start
thx for your help


" Obtain the solution of the following differential equation in the form of bessel equation;

$$x^2\frac{d^2 y}{dx^2} + \frac{1}{8}{x}\frac{dy}{dx} + (k^4x^8-6)y=0$$ "

 

[George Jones]

Is this question from a take-home exam, or is it a question that you couldn't do from an exam that you have already submitted?

 

[makasx]

yes I couldn't do this in exam, I try it in two way


* $$y= \sum_{n=\zero}^\infty C_nx^{(n+r)}}$$ from this I found $$\frac {dy}{dx} and \frac {d^2y}{dx^2}$$ and put them to equection and go on with frobenius method but I couldn't find "r" becouse of too many indicial equations so I couldn't find the method to recurrance equation and go on...


*I try to make the equation similar to $$x^2\frac{d^2 y}{dx^2} + {x}\frac{dy}{dx} + (\beta x^2 - n^2)y=0$$ so then I could write $$y(x) = AJ_{n}(\beta x) + BY_{n}(\beta x)$$ is the solution;

I try to put $$y=x^\alpha t$$ also $$t=x^\alpha with \frac {dy}{dx}=\frac {dy}{dt}*\frac{dt}{dx}$$ ,but couldn'tmake it similar

can you give me a way to strat

 

[muzialis]

Hi there,

have a look at the pdf, it might help

All the Best

Muzialis

 

[makasx]

I found some example questions with reducing equations and answers, also find my problem's reducing equations;(from KREYSZIG -advanced engineering mathematics)
thx

http://img41.imageshack.us/img41/43/exampless.jpg