Applications of differential equations

Modelling with differential equations, including exponential growth and decay, motion with air resistance and terminal velocity, mixing problems, the spread of a rumour or disease with the logistic model, and interpreting solutions.

Common models

Situation	DE	Behaviour
Rate proportional to the amount	$\dfrac{dN}{dt} = kN$	Exponential growth or decay
Rate proportional to the difference from a fixed value	$\dfrac{dN}{dt} = k(N - P)$	Approaches $P$ (for $k \lt 0$)
Growth limited by a capacity	$\dfrac{dP}{dt} = kP\left(1 - \dfrac{P}{K}\right)$	S-shaped, approaches $K$

Motion with air resistance

Example 1●●●

A skydiver falls from rest, and $\dfrac{dv}{dt} = 10 - 0.5v$. Find $v$ as a function of $t$ and the terminal velocity.

$\dfrac{dv}{dt} = -0.5(v - 20)$, so $v = 20 + Ae^{-0.5t}$.

At $t = 0$, $v = 0$, so $A = -20$ and $v = 20(1 - e^{-0.5t})$.

As $t \to \infty$, $v \to 20$ m/s. This is the terminal velocity, where air resistance balances gravity.

Mixing problems

Example 2●●●

A tank holds $100$ L of pure water. Brine with $2$ g/L of salt flows in at $5$ L/min, and the mixture flows out at $5$ L/min. Find the amount of salt $S$ after $t$ minutes.

Rate in: $2 \times 5 = 10$ g/min. Rate out: $\dfrac{S}{100} \times 5 = \dfrac{S}{20}$ g/min.

$\dfrac{dS}{dt} = 10 - \dfrac{S}{20} = -\dfrac{1}{20}(S - 200)$, so $S = 200 + Ae^{-t/20}$.

$S(0) = 0$ gives $A = -200$, so $S = 200\big(1 - e^{-t/20}\big)$. In the long run, $S \to 200$ g, the salt in $100$ L at $2$ g/L.

Spread of a rumour

Example 3●●

A rumour spreads through a school of $800$ students so that $\dfrac{dN}{dt} = 0.001N(800 - N)$. When is it spreading fastest?

$\dfrac{dN}{dt} = 0.8N\left(1 - \dfrac{N}{800}\right)$, a logistic equation with $K = 800$.

It spreads fastest when $N = 400$, half the school.

Flashcards

Terminal velocity

The limit of $v$ as $t \to \infty$, where $\dfrac{dv}{dt} = 0$.

Mixing problem

$\dfrac{dS}{dt} = \text{rate in} - \text{rate out}$

Rate out in a mixing problem

Concentration $\times$ outflow rate

Practice

For $\dfrac{dv}{dt} = 10 - 0.5v$, find the terminal velocity.
In Example 1, find $v$ when $t = 2\ln 2$.
In Example 2, how much salt is in the tank in the long run, in grams?
In Example 2, find $S$ when $t = 20\ln 2$.
In Example 3, how many students know the rumour when it spreads fastest?
A disease spreads so that $N = \dfrac{1000}{1 + 999e^{-t}}$. How many people are infected at $t = 0$?
"The rate of decay is proportional to the amount present" gives:
In a mixing problem, the outflow rate of salt depends on: