(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Hello guys! I've never dealt with an ODE having 2 singularities at once, I tried to solve it but ran out of ideas. I must solve ##(x-2)y''+3y'+4\frac{y}{x^2}=0##.

2. Relevant equations

Not sure.

3. The attempt at a solution

I rewrote the ODE into the form ##y''+\frac{3}{x-2}y'+4\frac{y}{x^2(x-2)}=0##. I notice that the singularities at ##x=2## and ##x=0## are both regular, so that Frobenius method should find at least 1 solution around any of these singularities.

So I first tried to expand the solution around ##x=2## first. Seeking solution(s) of the form ##\phi (x)=\sum _{n=0}^\infty a_n (x-2)^{n+c}##, I reached that [tex]\sum _{n=0}^\infty a_n(n+c)(n+c-1) (x-2)^{n+c-2}+ 3 \sum _{n=0}^\infty a_n (n+c) (x-2)^{n+c-2}+\frac{4}{x^2}\sum _{n=0}^\infty a_n (x-2)^{n+c}=0[/tex]. I stopped right there, because of the "1/x²" factor. But now that I think, maybe I can just "get rid of it" and it won't affect the solution of the ODE if I simply throw it away? Because that equation is satisfied for any x, so I guess this is enough of a reason to get rid of it?

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# Homework Help: Second order ODE, I think 2 regular points

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