IAL 2022 Jan Q8 | 國際數學站

(a) Show that the transformation $v = y - 2x$ transforms the differential equation

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

into the differential equation

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

(4)

(b) Solve the differential equation (II) to determine $v$ as a function of $x$

(4)

(1)

(d) Sketch the solution curve that passes through the point $(-1, -1)$ .

On your sketch show clearly the equation of any horizontal or vertical asymptotes.

You do not need to find the coordinates of any intercepts with the coordinate axes or the coordinates of any stationary points.

(5)