WAEC 2010 · Paper 2 · Q10

(a)
If $x^2 + y^2 = py(1 + x^2)$ , where $p$ is a constant, find $\dfrac{dy}{dx}$ .
(b)
A curve cuts the $x$ -axis at the points where $x = 0$ and $x = 3$ . If $\dfrac{d^2y}{dx^2} = 4(3 - x)^2$ , find the equation of the curve.

Worked solution (try it first)

Differentiate each side with respect to $x$ , using the product rule on the right: $2x + 2y\dfrac{dy}{dx} = p(1 + x^2)\dfrac{dy}{dx} + 2pxy$ .
Collect the $\dfrac{dy}{dx}$ terms: $(2y - p - px^2)\dfrac{dy}{dx} = 2pxy - 2x$ .
Divide: $\dfrac{dy}{dx} = \dfrac{2pxy - 2x}{2y - p - px^2}$ .