Homogeneous Differential Equations (DP IB Analysis & Approaches (AA)): Revision Note

Homogeneous differential equations

What is a homogeneous first order differential equation?

• If a first order differential equation can be written in the form  then it is said to be homogeneous

  • I.e. is equal to a function of

How do I solve a homogeneous first order differential equation?

• These equations can be solved using the substitution 

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

• STEP 2
  Replace all instances of  in your equation with v

• STEP 3
  Use the product rule and implicit differentiation to replace in your equation with 

  • This is because

• STEP 4
  Solve your new differential equation to find the solution in terms of v and x

  • You may need to use other methods for differential equations, such as separation of variables, at this stage

• STEP 5
  Substitute  into the solution from Step 4, in order to find the solution in terms of y and x

What else should I know about solving homogeneous first order differential equations?

• After finding the solution in terms of y and x 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

  • Or you may be asked to express your answer in a particular form, such as y = f(x)

• It is sometimes possible to solve differential equations that are not homogeneous by using the substitution 

  • In an exam question, you would be told explicitly to use the substitution for such a situation

  • You would not be expected to know that you could use the substitution in a case where the differential equation is not homogeneous