Problem

$$\begin{align*} \frac{\mathrm{d}^2y}{\mathrm{d}x^2} = x^2y+\frac{\mathrm{d}y}{\mathrm{d}x} \end{align*}$$

(a) Show that

$$\begin{align*} \frac{\mathrm{d}^4y}{\mathrm{d}x^4} = \frac{\mathrm{d}^3y}{\mathrm{d}x^3} +Ax^2\frac{\mathrm{d}^2y}{\mathrm{d}x^2} +Bx\frac{\mathrm{d}y}{\mathrm{d}x} +Cy \end{align*}$$

where $A$, $B$ and $C$ are integers to be determined.

Given that $\dfrac{\mathrm{d}y}{\mathrm{d}x}=3$ and $y=1$ at $x=1$

(b) determine the Taylor series solution for $y$ in ascending powers of $(x-1)$, up to and including the term in $(x-1)^4$, giving each coefficient in simplest form.