Math 2565, Spring 2020

Modelling with Differential Equations

Sean Fitzpatrick
University of Lethbridge

Warm-Up

Solve the following linear differential equations:

  1. \(y'+3x^2y = \sin(x)e^{-x^3}\text{,}\) with \(y(0)=1\)

  2. \(\di \frac{1}{x^2+1}y'+xy=3\text{,}\) with \(y(0)=0\)

  3. \(xy'+4y=x^3-x\)

Quiz

Solve the differential equation

\begin{equation*} y'+3x^2y = x^2\text{.} \end{equation*}

Exponential growth and decay

Models phenomena where the rate of change is proportional to the current amount.

  • Bad for population, except maybe early on, for bacteria or something.

  • Great for radioactive (beta) decay.

Differential equation is

\begin{equation*} \frac{dN}{dt} = kN\text{,} \end{equation*}
where \(k\) can be positive (growth) or negative (decay). Typically write \(N_0 = N(0)\text{.}\)

Example

The newly-discovered element Lethbridgium has a half-life of 14 days.

How long will it take until only 10% of a sample remains?

Newton's Law of Cooling

Newton's equation is

\begin{equation*} \frac{dT}{dt} = k(A-T)\text{,} \end{equation*}
where \(A\) is the ambient temperature, and \(T\) is the temperature of some object.

Example:

A pot of soup has a temperature of \(95^\circ\) C. It takes 15 minutes to cool to \(80^\circ\) at a room temperature of \(20^\circ\text{.}\) How long would it take to cool if set outside on a day in February when it's \(-10^\circ\text{?}\)

(Ingore additional cooling effects due to the wind. Even if this is unreasonable in Lethbridge.)

Example

Class exercise: how does the solution to Newton's Law of Cooling change if the ambient temperature is not constant?

Find the general solution if \(A(t) = A_0\cos(\omega t)\text{.}\) (Assume \(A_0, \omega\gt 0\text{.}\))

Rate in, rate out

These problems typically involve predicting the amount of some substance in a water solution, where there is solution flowing in, and out, at different concentrations.

The main challenge is setting up the equation.

Example:

A 50 litre tank holds 25 litres of pure water. A salt solution with a concentration of 5 g/L is added to the tank at a rate of 2 L/min.

If the solution undergoes “perfect mixing” and is drained at a rate of 2 L/min, find the amount of salt in the tank as a function of time.

Solution

Example 1

Initially a tank contains 10 litres of pure water. Brine of unknown (but constant) concentration of salt is flowing in at 1 litre per minute. The water is mixed well and drained at 1 litre per minute. In 20 minutes there are 15 grams of salt in the tank. What is the concentration of salt in the incoming brine?

Example 2

Suppose a water tank is being pumped out at 3 L / min . The water tank starts at 10 L of clean water. Water with toxic substance is flowing into the tank at 2 L / min , with concentration \(20t\) g / L at time \(t\text{.}\) When the tank is half empty, how many grams of toxic substance are in the tank (assuming perfect mixing)?

Example 3

Suppose there are two lakes located on a stream. Clean water flows into the first lake, then the water from the first lake flows into the second lake, and then water from the second lake flows further downstream. The in and out flow from each lake is 500 litres per hour. The first lake contains 100 thousand litres of water and the second lake contains 200 thousand litres of water. A truck with 500 kg of toxic substance crashes into the first lake. Assume that the water is being continually mixed perfectly by the stream.

  1. Find the concentration of toxic substance as a function of time in both lakes.

  2. When will the concentration in the first lake be below 0.001 kg per litre?

  3. When will the concentration in the second lake be maximal?