Construct a series solution $y=y_{1}(x)$ valid in the neighbourhood of $x=0$, for the differential equation

$$\frac{d^{2} y}{d x^{2}}+4 x^{3} \frac{d y}{d x}+x^{2} y=0$$

satisfying

$$y_{1}=1, \frac{d y_{1}}{d x}=0 \quad \text { at } x=0 .$$

Find also a second solution $y=y_{2}(x)$ which satisfies

$$y_{2}=0, \frac{d y_{2}}{d x}=1 \quad \text { at } x=0 .$$

Obtain an expression for the Wronskian of these two solutions and show that

$$y_{2}(x)=y_{1}(x) \int_{0}^{x} \frac{e^{-\xi^{4}}}{y_{1}^{2}(\xi)} d \xi$$