the question says 'use the result to solve bessel's equation of index n=1/2'.
i wasn't sure what it means by index n=1/2?
Compute $\displaystyle \frac{dy}{dx}$ and $\displaystyle \frac{d^2y}{dx^2}$ in terms of $\displaystyle x$ and $\displaystyle \nu$ using the suggested $\displaystyle y(x)=x^{-1/2}\nu(x)$. It is labourious but it does give the quoted equation.
Now if we put $\displaystyle n=1/2$ in the final equation it reduces to:
$\displaystyle
\frac{d^2\nu}{dx^2}+\nu=0
$
which has general solution $\displaystyle \nu(x)=A\cos(x)+B\sin(x)$, so the solution of the Bessel equation with $\displaystyle n=1/2$ (that is of order $\displaystyle n=1/2$) is:
$\displaystyle
y(x)=x^{-1/2}[A\cos(x)+B\sin(x)]
$
RonL