Separation of Variables (DP IB Applications & Interpretation (AI)): Revision Note

Separation of Variables

What is separation of variables?

• Separation of variables can be used to solve certain types of first order differential equations

• Look out for equations of the form 

  • i.e.  is a function of multiplied by a function of

  • be careful – the ‘function of ’ may just be a constant!

    • For example in , and

• If the equation is in that form you can use separation of variables to try to solve it

• If the equation is not in that form you will need to use another solution method

How do I solve a differential equation using separation of variables?

• STEP 1: Rearrange the equation into the form 

• STEP 2: Take the integral of both sides to change the equation into the form 

• You can think of this step as ‘multiplying the across and integrating both sides’

  • Mathematically that’s not quite what is actually happening, but it will get you the right answer here!

• STEP 3: Work out the integrals on both sides of the equation to find the general solution to the differential equation

  • Don’t forget to include a constant of integration

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

• STEP 4: Use any boundary or initial conditions in the question to work out the value of the integration constant

• STEP 5: If necessary, rearrange the solution into the form required by the question