Separation of Variables (DP IB Maths: AI HL)

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Roger

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Roger

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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 fraction numerator d y over denominator d x end fraction equals g left parenthesis x right parenthesis h left parenthesis y right parenthesis
    • i.e. fraction numerator d y over denominator d x end fraction is a function of x multiplied by a function of y
    • be careful – the ‘function of xg left parenthesis x right parenthesis may just be a constant!
      • For example in fraction numerator d y over denominator d x end fraction equals 6 y, g left parenthesis x right parenthesis equals 6 and h left parenthesis y right parenthesis equals 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 open parentheses fraction numerator 1 over denominator h left parenthesis y right parenthesis end fraction close parentheses fraction numerator d y over denominator d x end fraction equals g left parenthesis x right parenthesis
  • STEP 2: Take the integral of both sides to change the equation into the form 

integral subscript blank superscript blank fraction numerator 1 over denominator h left parenthesis y right parenthesis end fraction space d y equals integral subscript blank superscript blank g left parenthesis x right parenthesis space d x

    • You can think of this step as ‘multiplying the d x 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

Examiner Tip

  • Be careful with letters – the equation on an exam may not use xand y as the variables
  • Unless the question asks for it, you don’t have to change your solution into y equals f left parenthesis x right parenthesis form – sometimes it might be more convenient to leave your solution in another form

Worked example

For each of the following differential equations, either (i) solve the equation by using separation of variables giving your answer in the form y equals f left parenthesis x right parenthesis, or (ii) state why the equation may not be solved using separation of variables.

a)       fraction numerator d y over denominator d x end fraction equals fraction numerator straight e to the power of x plus 4 x over denominator 3 y squared end fraction.

5-10-2-ib-aa-hl-separation-of-variables-a-we-solution

b)       fraction numerator d y over denominator d x end fraction equals 4 x y minus 2 ln space x.

5-10-2-ib-aa-hl-separation-of-variables-b-we-solution

c)       fraction numerator d y over denominator d x end fraction equals 3 y, given that y equals 2 when x equals 0.

5-6-2-ib-ai-hl-separation-of-variables-c-we-solution

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Roger

Author: Roger

Expertise: Maths

Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.