Second Order Differential Equations (DP IB Applications & Interpretation (AI)): Revision Note

Euler's method: second order

What is a second order differential equation?

• A second order differential equation is a differential equation containing one or more second derivatives

• In this course you consider second order differential equations of the form

  

• Some examples include;

  • 

  • 

How do I apply Euler’s method to second order differential equations?

• You need to turn the second order differential equation into a pair of coupled first order differential equations

• Use the substitution

  • Then 

• The pair of equations then becomes the system

• You can then use Euler's method in the same way

• 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

• 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 

Exact solutions of second order differential equations

How can I find the exact solution for a second order differential equation?

• You might be asked to find exact solutions to second order differential equations of the form
  

  
  

  •  where and are constants

• Use the substitution to turn it into a coupled differential equation

• This can be written in the form 

  • 

  • 

  • 

• You can then investigate the solutions of the coupled differential equations

  • The characteristic equation is

• If the eigenvalues and are real, distinct, non-zero

  • The exact solution is of the form

    • Where and are constants

  • You can find the value of the constants given initial or boundary conditions

    • Remember that

    • This helps if you are given the initial value of

• For example, consider

  • The eigenvalues of are 1 and 3

  • The general solution is