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

Euler's Method: Second Order

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

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

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

  

  • You may need to rearrange the differential equation given to get it in this form

• In order to apply Euler’s method, use the substitution  to turn the second order differential equation into a pair of coupled first order differential equations

  • If , then 

  • This changes the second order differential equation into the coupled system

• Approximate solutions to this coupled system can then be found using the standard Euler’s method for coupled systems

  • See the notes on this method in the revision note 5.6.4 Approximate Solutions to Differential Equations

Exact Solutions & Phase Portraits: Second Order

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

• In some cases we can apply methods we already know to find the exact solutions for second order differential equations

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

  
  

  •  are constants

• Use the substitution  to turn the second order differential equation into a pair of coupled first order differential equations

  • If , then 

  • This changes the second order differential equation into the coupled system
    

    
    

  • The coupled system may also be represented in matrix form as

    

    • In the ‘dot notation’ here  and

  • That can be written even more succinctly as 

    • Here , , and 

• Once the original equation has been rewritten in matrix form, the standard method for finding exact solutions of systems of coupled differential equations may be used

  • The solutions will depend on the eigenvalues and eigenvectors of the matrix M

  • For the details of the solution method see the revision note 5.7.1 Coupled Differential Equations

  • Remember that exam questions will only ask for exact solutions for cases where the eigenvalues of M are real and distinct

How can I use phase portraits to investigate the solutions to second order differential equations?

• Here we are again considering second order differential equations of the form

  
  

  • a & b are real constants

• As shown above, the substitution can be used to convert this second order differential equation into a system of coupled first order differential equations of the form 

  • Here , , and 

• Once the equation has been rewritten in this form, you may use the standard methods to construct a phase portrait or sketch a solution trajectory for the equation

  • For the details of the phase portrait and solution trajectory methods see the revision note 5.7.1 Coupled Differential Equations

  • When interpreting a phase portrait or solution trajectory sketch, don’t forget that 

    • So if x represents the displacement of a particle, for example, then  will represent the particle’s velocity