Above is the question, Below is how I chose to attack it.
My approach is to compute the derivative and second derivatives of J2, and then substitute them into the given differential equation to prove that J2 is indeed a power series solution. However, during computation, I found that J2 is not a power series solution. Have I done something wrong in my computation, especially when I shifted index, or was the question incorrect?
I think that the Bessel Function of the first kind you are given is simply wrong: $$J_2(x)=\sum_{k=0}^\infty \frac {(-1)^k}{k!(\color {red}{k+2})!2^{2k+2}}x^{2k+2}$$
Where you have a factorial of $k+3$ at the denominator. The correct general formula is: $$J_n(x)=\sum_{k=0}^\infty \frac {(-1)^k}{k!({k+n})!2^{2k+n}}x^{2k+n}$$