Mathematics 9709 · AS & A Level · Differential equations
Differential equations — practice question
For the curve drawn in the diagram, the normal to the curve at the point $P$ with coordinates $(x, y)$ intersects the $x$-axis at $N$. Point $M$ is the perpendicular drop from $P$ to the $x$-axis. The curve is defined so that, for every value of $x$ in the range $0 \leq x < \frac{1}{2}\pi$, the area of triangle $PMN$ is $\tan x$.
(a(i))[1]
Show that $\frac{MN}{y} = \frac{dy}{dx}$.
(a(ii))[2]
Hence show that $x$ and $y$ satisfy the differential equation $\frac{1}{2}y^2 \frac{dy}{dx} = \tan x$.
(b)[6]
Given that $y = 1$ when $x = 0$, solve this differential equation to determine the equation of the curve, with $y$ expressed in terms of $x$.
Worked solution & mark scheme
This 9-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Justify the stated result $\frac{MN}{y}=\frac{dy}{dx}$” …