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IBMathsDPAnalysis & Approaches (AA)HLRevision Notes5. CalculusDifferential EquationsEuler's Method

Euler's Method (DP IB Analysis & Approaches (AA)): Revision Note

Roger B

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Roger B

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

Euler’s method: first order

What is Euler’s method?

  • Euler’s method is a numerical method for finding approximate solutions to differential equations

  • It treats the derivatives in the equation as being constant over short ‘steps’

  • The accuracy of the Euler’s Method approximation can be improved by making the step sizes smaller

How do I use Euler’s method with a first order differential equation?

  • STEP 1: Make sure your differential equation is in fraction numerator straight d y over denominator straight d x end fraction equals f open parentheses x comma blank y close parentheses form

  • STEP 2: Write down the recursion equations using the formulae y subscript n plus 1 end subscript equals y subscript n plus h cross times f open parentheses x subscript n comma blank y subscript n close parentheses and x subscript n plus 1 end subscript equals x subscript n plus h from the exam formula booklet

    • h in those equations is the step size

    • the exam question will usually tell you the correct value of h to use 

  • STEP 3: Use the recursion feature on your GDC to calculate the Euler’s method approximation over the correct number of steps

    • the values for x subscript 0 and y subscript 0 will come from the boundary conditions given in the question

Examiner Tips and Tricks

  • Be careful with letters – in the equations in the exam, and in your GDC’s recursion calculator, the variables may not be x and y

  • If an exam question asks you how to improve an Euler’s method approximation, the answer will almost always have to do with decreasing the step size!

Worked Example

Consider the differential equation fraction numerator straight d y over denominator straight d x end fraction plus y equals x plus 1 with the boundary condition space y left parenthesis 0 right parenthesis equals 0.5.

a) Apply Euler’s method with a step size of h equals 0.2 to approximate the solution to the differential equation at x equals 1.

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b) Explain how the accuracy of the approximation in part (a) could be improved.

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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.