Modelling with Differential Equations

Why are differential equations used to model real-world situations?

A differential equation is an equation that contains one or more derivatives
Derivatives deal with rates of change, and with the way that variables change with respect to one another
Therefore differential equations are a natural way to model real-world situations involving change

Most frequently in real-world situations we are interested in how things change over time, so the derivatives used will usually be with respect to time t

How do I set up a differential equation to model a situation?

An exam question may require you to create a differential equation from information provided
The question will provide a context from which the differential equation is to be created
Most often this will involve the rate of change of a variable being proportional to some function of the variable

For example, the rate of change of a population of bacteria, P, at a particular time may be proportional to the size of the population at that time

The expression ‘rate of’ (‘rate of change of…’, ‘rate of growth of…’, etc.) in a modelling question is a strong hint that a differential equation is needed, involving derivatives with respect to time t

So with the bacteria example above, the equation will involve the derivative $\frac{d P}{d t}$

Recall the basic equation of proportionality

If y is proportional to x, then y = kx for some constant of proportionality k

So for the bacteria example above the differential equation needed would be $\frac{d P}{d t} = k P$

The precise value of k will generally not be known at the start, but will need to be found as part of the process of solving the differential equation
It can often be useful to assume that k > 0 when setting up your equation

In this case, -k will be used in the differential equation in situations where the rate of change is expected to be negative
So in the bacteria example, if it were known that the population of bacteria was decreasing, then the equation could instead be written $\frac{d P}{d t} = - k P$

Worked example

In a particular pond, the rate of change of the area covered by algae, A, at any time t is directly proportional to the square root of the area covered by algae at that time. Write down a differential equation to model this situation.

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Newton’s Law of Cooling states that the rate of change of the temperature of an object, T, at any time t is proportional to the difference between the temperature of the object and the ambient temperature of its surroundings, T_a , at that time. Assuming that the object starts off warmer than its surroundings, write down the differential equation implied by Newton’s Law of Cooling.

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Modelling with Differential Equations (DP IB Maths: AI HL)

Revision Note