Mathematics 9709 · AS & A Level · Differential equations
Differential equations — practice question
Let the number of micro-organisms in a population at time $t$ be represented by $M$. At any instant, the change in $M$ is assumed to obey the differential equation $\frac{dM}{dt} = k(\sqrt{M})\cos(0.02t)$, where $k$ is a constant and $M$ is treated as a continuous variable. It is given that when $t = 0$, $M = 100$.
(i)[5]
Solve the differential equation, obtaining a relation between $M$, $k$ and $t$.
(ii)[2]
Also given that $M = 196$ when $t = 50$, determine the value of $k$.
(iii)[2]
Obtain an expression for $M$ in terms of $t$ and determine the least possible number of micro-organisms.
Worked solution & mark scheme
This 9-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Separate the variables accurately and integrate one side” …