(a) The function $y(x)$ satisfies

$$y^{\prime \prime}+p(x) y^{\prime}+q(x) y=0$$

(i) Define the Wronskian $W(x)$ of two linearly independent solutions $y_{1}(x)$ and $y_{2}(x)$. Derive a linear first-order differential equation satisfied by $W(x)$.

(ii) Suppose that $y_{1}(x)$ is known. Use the Wronskian to write down a first-order differential equation for $y_{2}(x)$. Hence express $y_{2}(x)$ in terms of $y_{1}(x)$ and $W(x)$.

(b) Verify that $y_{1}(x)=\cos \left(x^{\gamma}\right)$ is a solution of

$$a x^{\alpha} y^{\prime \prime}+b x^{\alpha-1} y^{\prime}+y=0,$$

where $a, b, \alpha$ and $\gamma$ are constants, provided that these constants satisfy certain conditions which you should determine.

Use the method that you described in part (a) to find a solution which is linearly independent of $y_{1}(x)$.