Find coefficients $a_n$, n $\subset$ N Such that the function f given by :
$f(x)$=$\sum_{n=0}^\infty a_nx^n$
Statisfy the equation: $f''(x)$+$x^2f'(x)$+$xf(x)$=0, $f(0)$=1, $f'(0)$=1
Give $a_0...,a_3$ explicit. Determine the radius of convergence.
Now i did the algebra regarding the power series and ended up with
$\sum_{n=0}^\infty (n+1)(n+2) a_{n+2}x^n+\sum_{n=2}^\infty (n-1)a_{n-1}x^n+\sum_{n=1}^\infty a_{n-1}x^n$ = 0
And now i was trying to find the coefficients by using number plug in for:
$n=0$: (1)(2)$a_2$=0 so $a_2$=0 For $n=1$: (2)(3)$a_3$+$a_0$=0 so now i dont know what i should do next? Should i use the fact that $f(0)=1$ so that means that $a_0$=1? Then it follows that $a_3$=-1/6 but then i know that the Taylor series will be the function $sin(x)$ and that will start with 0 so $a_0$=0. What am i doing wrong? Please help..