Let $y_{1}$ and $y_{2}$ be two linearly independent solutions to the differential equation

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

Show that the Wronskian $W=y_{1} y_{2}^{\prime}-y_{2} y_{1}^{\prime}$ satisfies

$$\frac{\mathrm{d} W}{\mathrm{~d} x}+p W=0 .$$

Deduce that if $y_{2}\left(x_{0}\right)=0$ then

$$y_{2}(x)=y_{1}(x) \int_{x_{0}}^{x} \frac{W(t)}{y_{1}(t)^{2}} \mathrm{~d} t .$$

Given that $y_{1}(x)=x^{3}$ satisfies the equation

$$x^{2} \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}-x \frac{\mathrm{d} y}{\mathrm{~d} x}-3 y=0$$

find the solution which satisfies $y(1)=0$ and $y^{\prime}(1)=1$.