IAL 2022 Oct Q1

Given that

(a) Express $\frac{\mathrm{d}^3y}{\mathrm{d}x^3}$ in terms of $x$ , $\frac{\mathrm{d}y}{\mathrm{d}x}$ and $\frac{\mathrm{d}^2y}{\mathrm{d}x^2}$

At $x = 0$ , $y = 2$ and $\frac{\mathrm{d}y}{\mathrm{d}x} = 5$

(b) Determine the value of $\frac{\mathrm{d}^3y}{\mathrm{d}x^3}$ at $x = 0$

(c) Express $y$ as a series in ascending powers of $x$ , up to and including the term in $x^3$