The linear second-order differential equation

$$\frac{d^{2} y}{d x^{2}}+p(x) \frac{d y}{d x}+q(x) y=0$$

has linearly independent solutions $y_{1}(x)$ and $y_{2}(x)$. Define the Wronskian $W$ of $y_{1}(x)$ and $y_{2}(x)$.

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 $W$.

Show further that $W$ satisfies the differential equation

$$\frac{d W}{d x}+p(x) W=0$$

Verify that $y_{1}(x)=x^{2}-2 x+1$ is a solution of

$$(x-1)^{2} \frac{d^{2} y}{d x^{2}}+(x-1) \frac{d y}{d x}-4 y=0 .$$

Compute the Wronskian and hence determine a second, linearly independent, solution of $(*)$.