- #1
fluidistic
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Homework Statement
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##.
Homework Equations
Not sure.
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?