Question

I have one question about Bessel's equation. I'll write in comment

Answer 1

I have to find solution of equation
\[x ^{2}y''+xy'+x ^{2}y=0\]

I found that \[y _{1}=\sum_{0}^{\infty} (-1)^{k}\frac{ a _{0} }{ 2^{2k}(k!)^{2} }x ^{2k}\]

Then I found, for \[y _{2}\]

\[a _{0}(r)r ^{2}x ^{r}=0\]
\[a _{1}(r)(r+1)^{2}x ^{r+1}=0\]
\[a _{n}(r)=-\frac{ a _{n-2}(r) }{ (n+r)^{2} }\]

Now I don't know how to choose coefficients. How do I determine, for example, if
\[a _{0}=0\] or r^2=0?

Answer 2

Since I got that \[r _{1}=r _{2}=0\], is that how do I know that r=0 and not a0=0
and from that, (r+1)^2 is not 0, that is r1 and r2 are not -1, and than I choose a1=0?