Mathematics 9709 · AS & A Level · Differential equations
Differential equations — practice question
A fungal disease is now affecting some of the trees in a forest. Let $x$ be the fraction of trees affected after $t$ years. The rate of increase of $x$ is proportional to the product of the fraction already affected and the fraction still unaffected.
(a)[1]
Explain why, after $t$ years, $\frac{dx}{dt} = kx(1-x)$, where $k$ is a constant.
(b)[8]
At the moment the disease is first noticed, one quarter of the trees are affected. After two years, the fraction affected is one third.
Solve the differential equation to determine how many years elapse from the time when the disease is first detected to the point when three quarters of the trees are affected. Give your answer to the nearest year.
Worked solution & mark scheme
This 9-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Obtain $\frac{dx}{dt}=kx(1-x)$ and indicate where the factors come from.” …