IAL 2022 June Q5 | 國際數學站

Given that

\begin{align*} y \frac{\mathrm{d}^2y}{\mathrm{d}x^2} + 2 \left( \frac{\mathrm{d}y}{\mathrm{d}x} \right)^2 - 2y =\,& 0 \qquad y > 0\\[2mm] \end{align*}

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

(4)

Given that $y = 2$ and $\frac{\mathrm{d}y}{\mathrm{d}x} = 1$ at $x = 0$

(b) determine a series solution for $y$ in ascending powers of $x$ , up to and including the term in $x^3$ , giving each coefficient in its simplest form.

(4)