Integrating Factor (DP IB Analysis & Approaches (AA)): Revision Note

Integrating factor

What is an integrating factor?

• An integrating factor can be used to solve a differential equation that can be written in the standard form 

• For an equation in standard form, the integrating factor is 

How do I use an integrating factor to solve a differential equation?

• STEP 1
  If necessary, rearrange the differential equation into standard form
   

• STEP 2
  Find the integrating factor

  • Note that you don’t need to include a constant of integration here when you integrate 

• STEP 3
  Multiply both sides of the differential equation by the integrating factor

  • This will turn the equation into an exact differential equation of the form 

• STEP 4
  Integrate both sides of the equation with respect to x

  • The left side will automatically integrate to 

  • For the right side, integrate  using your usual techniques for integration

  • Don’t forget to include a constant of integration

    • Although there are two integrals, you only need to include one constant of integration

• STEP 5
  Rearrange your solution to get it in the form y = f(x)

What else should I know about using an integrating factor to solve differential equations?

• After finding the general solution using the steps above you may be asked to do other things with the solution

  • For example you may be asked to find the solution corresponding to certain initial or boundary conditions