Approximate Solutions to Differential Equations using Euler's Method (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 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 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