Mathematics 9709 · AS & A Level · Differential equations
Differential equations — practice question
Let $x$ denote the population size at time $t$. If $x$ is treated as a continuous variable, it satisfies the differential equation $\frac{dx}{dt} = \frac{xe^{-t}}{k + e^{-t}}$, where $k$ is a positive constant.
(a)[6]
With $x = 10$ when $t = 0$, solve the differential equation to produce a relation connecting $x$, $k$ and $t$.
(b)[2]
Also given that $x = 20$ when $t = 1$, show that $k = 1 - \frac{2}{e}$.
(c)[2]
Show that the number of organisms does not reach $48$, no matter how large $t$ becomes.
Worked solution & mark scheme
This 10-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Carry out separation of variables correctly and make an attempt to integrate one side” …