Separation of Variables (College Board AP® Calculus AB)

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 equal to 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

  • then you can use separation of variables to try to solve it

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

• STEP 1
  Rearrange the equation into the form 

  • E.g.

    • 

• STEP 2
  Integrate both sides with respect to

  • This changes the equation into the form 

  • E.g.

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

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

• STEP 3
  Work out the integrals on both sides of the equation

  • Don’t forget to include a constant of integration

    • You only need one constant of integration, even though there are two integrals

  • E.g.

• STEP 4
  Rearrange the solution

  • E.g.

  • Note that you won't always be able to rewrite the solution in form

    • In this case is not correct, because the solutions also include the option

    • If an exam question requires the answer in a particular form, be sure to rearrange into that form

  • Also note that is just another arbitrary integration constant

    • So would be a 'neater' way to write the solution

• This method gives the general solution to the differential equation

  • For finding the particular solution, see the 'Particular Solutions' study guide