Use the change of variable $z=\sin ^{2} x$, to rewrite the equation

$$\frac{d^{2} y}{d x^{2}}+k^{2} y=0$$

where $k$ is a real non-zero number, as the hypergeometric equation

$$\frac{d^{2} w}{d z^{2}}+\left(\frac{C}{z}+\frac{1+A+B-C}{z-1}\right) \frac{d w}{d z}+\frac{A B}{z(z-1)} w=0$$

where $y(x)=w(z)$, and $A, B$ and $C$ should be determined explicitly.

(i) Show that ( $\$)$ is a Papperitz equation, with 0,1 and $\infty$ as its regular singular points. Hence, write the corresponding Papperitz symbol,

$$P\left\{\begin{array}{cccc} 0 & 1 & \infty \\ 0 & 0 & A \\ 1-C & C-A-B & B \end{array}\right\}$$

in terms of $k$.

(ii) By solving ( $\dagger$ ) directly or otherwise, find the hypergeometric function $F(A, B ; C ; z)$ that is the solution to $(\ddagger)$ and is analytic at $z=0$ corresponding to the exponent 0 at $z=0$, and satisfies $F(A, B ; C ; 0)=1$; moreover, write it in terms of $k$ and

(iii) By performing a suitable exponential shifting find the second solution, independent of $F(A, B ; C ; z)$, which corresponds to the exponent $1-C$, and hence write $F\left(\frac{1+k}{2}, \frac{1-k}{2} ; \frac{3}{2} ; z\right)$ in terms of $k$ and $x$.