For positive n, the ordinary differential equation
$$y''+\dfrac{1}{x}y'+\left(1-\dfrac{n^2}{x^2}\right)y=0$$
has as a solution the Bessel function of order n, $J_n\left(x\right)=x^n\sum^{\infty}_{k=0}a_kx^k$
Determine $a_1$ by substituting the series for the Bessel function into the differential equation.
So I have $y(x)=\sum_{k=0}^{\infty}a_kx^{k+r}$ , $y'(x)=\sum_{k=0}^{\infty}a_k(k+r)x^{k+r-1}$ and $y''(x)=\sum_{k=0}^{\infty}a_k(k+r)(k+r-1)x^{k+r-2}$
So subbing in I get:
$$\sum_{k=0}^{\infty}a_k(k+r)(k+r-1)x^{k+r-2}+\sum_{k=0}^{\infty}a_k(k+r)x^{k+r-2}+\sum_{k=0}^{\infty}a_kx^{k+r-2}=0$$
Then I get:
$$\sum_{k=2}^{\infty}a_k((k+r)(k+r-1)+(k+r)+1)x^{k+r-2}+[a_0(r(r-1)+r+1)x^{r-2}]+[a_1((r+1)r+(r+1)+1)x^{r-1}$$
I am unsure where to go from here