IAL 2021 Oct Q7 | 國際數學站

(a) Show that the transformation $x = t^2$ transforms the differential equation

\begin{align*} 4x \frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + 2(1 + 2\sqrt{x}) \frac{\mathrm{d}y}{\mathrm{d}x} - 15y =\,& 15x \qquad \text{(I)}\\[2mm] \end{align*}

into the differential equation

\begin{align*} \frac{\mathrm{d}^2 y}{\mathrm{d}t^2} + 2\frac{\mathrm{d}y}{\mathrm{d}t} - 15y =\,& 15t^2 \qquad \text{(II)}\\[2mm] \end{align*}

(5)

(b) Solve differential equation (II) to determine $y$ in terms of $t$ .

(5)

(1)