Bessel in AP?

$2n J_{n}=x(J_{n-1}+J_{n+1})$

$\large{\frac{x}{2} J_{n-1}=n J_{n}-\frac{x}{2} J_{n+1}}\quad \quad \quad (1)$

$n \rightarrow n+2$ in $(1)$

$\large{\frac{x}{2} J_{n+1}=(n+2) J_{n+2}-\frac{x}{2} J_{n+3}}$

Put the value of $\large{\frac{x}{2} J_{n+1}}$ in $(1)$

We get $\large{\frac{x}{2} J_{n-1}=n J_{n}-(n+2)J_{n+2}+\frac{x}{2} J_{n+3}}\quad \quad \quad (2)$

Now $n \rightarrow n+4$ in $(1)$

Find value of $\large{\frac{x}{2} J_{n+3}}$ and place it in $(2)$

On repeating these steps we finally get

$\large{\frac{x}{2} J_{n-1}=n J_{n}-(n+2) J_{n+2}+(n+4) J_{n+4}-........}$

In this case $n=2$

Thus we are left with

$\large{\frac{1}{2} \int x^2 J_{1} dx=\frac{x^2}{2}J_{2}+c}$