Mathematics 9709 · AS & A Level · Differential equations
Differential equations — practice question
A plantation with total area $20\text{ km}^2$ is being affected by a plant disease. At time $t$ years, the diseased area is $x\text{ km}^2$, and the rate at which $x$ increases is proportional to the ratio of the infected area to the area that is still unaffected. When $t = 0$, $x = 1$ and $\frac{dx}{dt} = 1$.
(a)[2]
Show that the variables $x$ and $t$ satisfy the differential equation $\frac{dx}{dt} = \frac{19x}{20 - x}$.
(b)[5]
Solve the differential equation and show that, when $t = 1$, the value of $x$ satisfies $x = e^{0.9 + 0.05x}$.
(c)[3]
Apply an iterative method based on the equation in part (b), starting from $2$, to find $x$ correct to $2$ decimal places. Record each iteration to $4$ decimal places.
(d)[1]
Calculate the time $t$ when the entire plantation is infected.
Worked solution & mark scheme
This 11-mark question has a full step-by-step worked solution and mark scheme. One marking point: “State or imply an equation in the form $\frac{dx}{dt}=k-\frac{x}{20-x}$” …