Approximate Solutions to Differential Equations (DP IB Applications & Interpretation (AI)): Revision Note

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  form

• STEP 2: Write down the recursion equations using the formulae  and  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  and  will come from the boundary conditions given in the question

Euler's Method: Coupled Systems

How do I use Euler’s method with coupled first order differential equations?

• STEP 1: Make sure your coupled differential equations are in  and  form

• STEP 2: Write down the recursion equations using the formulae ,  and  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 ,  and  will come from the boundary conditions given in the question

  • frequently you will be given an initial condition

    • look out for terms like ‘initially’ or ‘at the start’

    • in this case