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IBMathsDPAnalysis & Approaches (AA)HLRevision Notes5. CalculusDifferential EquationsModelling with Differential Equations

Modelling with Differential Equations (DP IB Analysis & Approaches (AA)): Revision Note

Roger B

Written by: Roger B

Reviewed by: Dan Finlay

Updated on

DP Analysis & Approaches (AA): HLDP Applications & Interpretation (AI): HLDP Analysis & Approaches (AA): HL

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

    • In real-world situations we are usually 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 to create the differential equation

  • This will usually 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  fraction numerator d P over denominator d t end fraction

  • 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 fraction numerator d P over denominator d t end fraction equals k P

    • The precise value of k will generally not be known at the start

      • You will need to be find it as part of the process of solving the differential equation

Examiner Tips and Tricks

It is often useful to assume that k greater than 0 when setting up your differential equation. In this case, negative k will be used in situations where the rate of change is expected to be negative (i.e. where the quantity is decreasing).

So in the bacteria example, if you know that the population of bacteria is decreasing, then the equation could instead be written fraction numerator d P over denominator d t end fraction equals negative k P, where k greater than 0.

Worked Example

a) 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.

5-10-3-ib-aa-hl-modelling-with-diff-eqns-a-we-solution

b) 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, Ta , 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.

5-10-3-ib-aa-hl-modelling-with-diff-eqns-b-we-solution

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    Roger B

    Author: Roger B

    Expertise: Maths Content Creator

    Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.