Mathematics 9709 · AS & A Level · Differential equations
Differential equations — practice question
At time $t$ years, the population of a country is $N$ millions. It is assumed that, at any instant, the rate at which $N$ increases is proportional to the product of $N$ and $(1 - 0.01N)$. When $t = 0$, $N = 20$ and $\frac{dN}{dt} = 0.32$.
(i)[1]
With $N$ and $t$ regarded as continuous variables, show that they satisfy $\frac{dN}{dt} = 0.02N(1 - 0.01N)$.
(ii)[8]
Solve the differential equation to obtain $t$ written in terms of $N$.
(iii)[1]
Determine the time when the population has become double its value at $t = 0$.
Worked solution & mark scheme
This 10-mark question has a full step-by-step worked solution and mark scheme. One marking point: “State or imply $\frac{dN}{dt}=kN(1-0.01N)$ and deduce $k=0.02$” …