Numerical Solutions to Differential Equations (DP IB Analysis & Approaches (AA)) : Revision Note

First Order Differential Equations

What is a differential equation?

• A differential equation is simply an equation that contains derivatives

  • For example  is a differential equation

  • And so is 

What is a first order differential equation?

• A first order differential equation is a differential equation that contains first derivatives but no second (or higher) derivatives

  • For example  is a first order differential equation

  • But  is not a first order differential equation, because it contains the second derivative 

Wait – haven’t I seen first order differential equations before?

• Yes you have!

  • For example  is also a first order differential equation, because it contains a first derivative and no second (or higher) derivatives

  • But for that equation you can just integrate to find the solution y = x³ + c (where c is a constant of integration)

• In this section of the course you learn how to solve differential equations that can’t just be solved right away by integrating

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