Separation of Variables (DP IB Maths: AI HL)

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