Derive the solution y2(x) = J-ν(x) using the Frobenius method for Bessel's differential equation x2y'' + xy' + (x2 - ν2)y = 0.

To derive the solution $y_2(x) = J_{-\nu}(x)$ using the Frobenius method for Bessel's differential equation $x^2 y'' + xy' + (x^2 - \nu^2) y = 0$:

Step 1: Assume a Frobenius series solution.
Let $y(x) = \sum_{n=0}^{\infty} a_n x^{n+r}$.
Then the first and second derivatives are:

$$y'(x) = \sum_{n=0}^{\infty} a_n (n+r) x^{n+r-1}$$ $$$$