IAL 2022 Oct Q8 | 國際數學站

(a) Show that the transformation $x = \mathrm{e}^u$ transforms the differential equation

\begin{align*} x^2\frac{\mathrm{d}^2y}{\mathrm{d}x^2} + 3x\frac{\mathrm{d}y}{\mathrm{d}x} - 8y =\,& 4\ln x \qquad x > 0 \qquad \text{(I)}\\[2mm] \end{align*}

into the differential equation

\begin{align*} \frac{\mathrm{d}^2y}{\mathrm{d}u^2} + 2\frac{\mathrm{d}y}{\mathrm{d}u} - 8y =\,& 4u \qquad \text{(II)}\\[2mm] \end{align*}

(6)

(b) Determine the general solution of differential equation (II), expressing $y$ as a function of $u$ .

(7)

(1)