Separation of Variables (DP IB Analysis & Approaches (AA): HL): Revision Note

Separation of variables

What is separation of variables?

• Separation of variables is a method that can be used to solve certain types of first order differential equations

• Look out for equations of the form $\frac{dy}{dx}=g\left(x\right)h\left(y\right)$

  • i.e. $\frac{dy}{dx}$ is a function of $\mathit{x}$ multiplied by a function of $\mathit{y}$

• 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 $\left(\frac{1}{h\left(y\right)}\right)\frac{dy}{dx}=g\left(x\right)$
   

• STEP 2
  Take the integral of both sides to change the equation into the form
  
  $\int \frac{1}{h\left(y\right)}dy=\int g\left(x\right)dx$
   

  • You can think of this step as ‘multiplying the $dx$ 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

  • Look out for integrals that require you to use partial fractions to solve them

    • See the ‘Integrating with Partial Fractions’ revision note in Further 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