- #1
BearY
- 53
- 8
Homework Statement
Use Forbenius' method to solve this DE:
$$ 5x^2y''+xy'+(x^3-1)y=0$$
Homework Equations
Seek power series solution in the form ##y=\sum _{n=0}^{\infty } a_n x^{n+r}##, ##a_0\neq0##
The Attempt at a Solution
Sub in the ansatz y, get $$ \sum _{n=0}^{\infty }a_n[5(n+r)(n+r+1)+n+r-1]x^n + \sum _{n=0}^{\infty }a_nx^{n+r+3} = 0 $$
Align the powers of x, split first sum to combine with second sum:$$ \sum _{n=0}^{\infty }\{a_{n+3}[5(n+r+3)(n+r+2)+n+r+3-1]+a_n\}x^{n+3} + a_0[5r(r-1)+r-1]+a_1[r(r+1)r+r]x+a_2[5(r+2)(r+1)+r+1]x^2=0$$
This is where I get stuck, I am not sure about how to handle ##a_1## and ##a_2## term or how to find solutions of ##r## with them being there. I can't find out if ##a_1## and ##a_2## can be zero or not, whether or why it matters.
Thanks in advance.