IAL 2022 June Q7 | 國際數學站

(a) Show that the transformation $y = xv$ transforms the equation

\begin{align*} 3 \frac{\mathrm{d}^2y}{\mathrm{d}x^2} - \frac{6}{x} \frac{\mathrm{d}y}{\mathrm{d}x} + \frac{6y}{x^2} + 3y =\,& x^2 \qquad x \neq 0 \qquad \text{(I)}\\[2mm] \end{align*}

into the equation

\begin{align*} 3 \frac{\mathrm{d}^2v}{\mathrm{d}x^2} + 3v =\,& x \qquad \text{(II)}\\[2mm] \end{align*}

(6)

(b) Hence obtain the general solution of the differential equation (I), giving your answer in the form $y = \mathrm{f}(x)$

(6)