Solving ODEs w/ Frobenius Method: Q on b, c as Funcs of x

In summary, the conversation is about the Frobenius method for solving ODEs with the form (x^2)y'' + xby' + cy = 0 and the difference in using the method for equations with constant coefficients versus equations with variable coefficients. The conversation also touches on difficulties with applying the method and potential solutions such as using the substitution z=x^2y.
  • #1
I have been looking at the Frobenius method for solving ODEs of the form. I have a few questions on it.

(x^2)y'' + xby' + cy = 0

If b and c are functions of x, does one use the Frobenius method, where as if they are constants, it is an Euler Cauchy equation and you use y = x^r ??

Thats the first Q. anyway.

Thanks folks!:smile:
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  • #2
y=xr only works for the Euler-Cauchy equation, i.e. b and c are constants.

If b and c are functions of x, you cannot use the try function y=xr.
  • #3
Thanks for cledaring that up.

I'm having trouble applying the method. My textbook, (which I won't name but it's approach and exlanation in this section is absolutely terrible) isn't helping me much.

I have been trying to solve, for example,

xy'' + 5y' + xy = 0

So I get

SUM(n + r)(N + r -1)(a_n)x^(n + r -2) + SUM(5)(n + r)(a_n)x^(n + R -2) + SUM(a_n)x^(n + r +1) = 0

where SUM is the sum to infinity from n = 0.

and the general solution is of form y=(x^r)SUM(a_n)(x^n)

I don't know what to do know. The book's next steps are done without explanation really.
Can someone help me??
  • #4
x=0 is a regular singular point. There is at least one solution of the form y=(x^r)SUM(a_n)(x^n)
where r satisfy the indicial equation r2+4r=0.
r1=0 , r2=-4 and r1-r2 is an integer in this case.

The Frobenius method only guarantee for r=0 (the larger root) but not for r=-4 (but there is no harm for trying)

If you know anything about Bessel equation, I would suggest you solve the equation using the substitution z=x2y.

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