IAL 2021 June Q8 | 國際數學站

(a) Show that the substitution $v = y^{-2}$ transforms the differential equation

\begin{align*} \frac{\mathrm{d}y}{\mathrm{d}x} + 6xy = 3x e^{x^2} y^3 \qquad x > 0 \qquad \text{(I)} \end{align*}

into the differential equation

\begin{align*} \frac{\mathrm{d}v}{\mathrm{d}x} - 12vx = -6xe^{x^2} \qquad x > 0 \qquad \text{(II)} \end{align*}

(5)

(b) Hence find the general solution of the differential equation (I), giving your answer in the form $y^2 = f(x)$ .

(6)