Modelling with differential equations

​​In a nutshell

Differential equations have a wide variety of real-life applications and you may be expected to model a particular situation by constructing and solving a differential equation.

Example 1

A scientist is observing the number of animals in an enclosure. The scientist observes that the number of animals, $$N$$, is always half the rate of increase of the population over time.

i) Construct a differential equation and find a general solution to the number of animals at any time, $$t$$, where $$t$$ is the time in hours from when the scientist began their observation.

ii) Given that there were only $$50$$ animals when the scientist began their experiment, find an exact formula for the population $$N$$.

iii) Explain why this model is unrealistic.

Part i):

Form the differential equation and solve:

$$\begin{aligned}N&=\dfrac12 \times \dfrac {dN}{dt}\\ \dfrac{dN}{dt}&=2N\\ \dfrac{1}{N} \, dN &= 2 \, dt\\ \int \dfrac{1}{N} \, dN &= \int2 \, dt\\\ln \vert N\vert &=2t+C\\N&=Ae^{2t} \end{aligned}$$​​

The general solution is $$\underline{N=Ae^{2t}}$$.

​Part ii):

Use the boundary condition to find the value of $$A$$:

​$$\begin{aligned}t=0&,N=50\\N&=Ae^{2t}\\50&=Ae^{2(0)}\\50&=A \end{aligned}$$​​

The particular solution is $$\underline{N=50e^{2t}}$$.

Part iii):

The model is unrealistic as it assumes the number of animals will increase indefinitely over time, which is impossible.​

Note: When being asked to explain why a model is unrealistic, it is best to consider what will happen to the dependent variable if the independent variable is very large, or if the model fails to take into account various factors that can't be ignored in the real world.

Example 2

An empty, perfectly cylindrical container with radius $$8cm$$ is slowly filled with water. The rate of change of the height of the water is triple the time since the water started to enter the cylinder. Construct and solve a differential equation to find a formula for the volume $$V$$ of water as a function of time $$t$$.

Write down all the known information:

$$V=\pi (8)^2 h=64\pi h\Rightarrow \dfrac{dV}{dh}=64\pi \\ \dfrac{dh}{dt}=3t$$​​

Use the chain rule to find a formula for $$\dfrac{dV}{dt}$$ in terms of $$\dfrac{dV}{dh}$$ and $$\dfrac{dh}{dt}$$:

​$$\begin{aligned}\dfrac{dV}{dt}&=\dfrac{dV}{dh}\times \dfrac{dh}{dt}\\&=64\pi \times t\\&=64\pi t \end{aligned}$$​​

Solve the differential equation:

$$\dfrac{dV}{dt}=64\pi t \Rightarrow V(t)=32\pi t^2+C$$​​

The container was initially empty. Interpret this to give a boundary condition which can be used to find $$C$$:

​$$t=0,V=0\\ \begin{aligned}V(t)&=32\pi t^2+C\\0&=32\pi (0)^2+C\\C&=0 \end{aligned}$$​​

​$$\underline{V(t)=32\pi t^2}$$​​