Solving First Order Differential Equations (Edexcel A Level Further Maths: Core Pure)

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First Order Differential Equations

What is a differential equation?

  • A differential equation is simply an equation that contains derivatives
    • For example fraction numerator straight d y over denominator straight d x end fraction equals 12 x y squared is a differential equation
    • And so is fraction numerator straight d squared x over denominator straight d t squared end fraction minus 5 fraction numerator straight d x over denominator straight d t end fraction plus 7 x equals 5 sin invisible function application t

What is a first order differential equation?

  • A first order differential equation is a differential equation that contains first derivatives but no second (or higher) derivatives
    • For example fraction numerator straight d y over denominator straight d x end fraction equals 12 x y squared is a first order differential equation
    • But fraction numerator straight d squared x over denominator straight d t squared end fraction minus 5 fraction numerator straight d x over denominator straight d t end fraction plus 7 x equals 5 sin invisible function application t is not a first order differential equation, because it contains the second derivative fraction numerator straight d squared x over denominator straight d t squared end fraction
  • The general solution to a first order differential equation will have one unknown constant
  • To find the particular solution you will need to know an initial condition or a boundary condition

Wait – haven’t I seen first order differential equations before?

  • Yes you have!
    • For example fraction numerator straight d y over denominator straight d x end fraction equals 3 x squared is also a first order differential equation, because it contains a first derivative and no second (or higher) derivatives
    • But for that equation you can just integrate to find the solution y = x3 + c (where c is a constant of integration)
  • In A Level Maths you will have solved some first order differential equations using the method of separation of variables

Integrating Factors

What is an integrating factor?

  • An integrating factor can be used to solve a differential equation that can be written in the standard form fraction numerator d y over denominator d x end fraction plus p left parenthesis x right parenthesis y equals q left parenthesis x right parenthesis
    • Be careful – the ‘functions of xp(x) and q(x) may just be constants!
      • For example in fraction numerator d y over denominator d x end fraction plus 6 y equals straight e to the power of negative 2 x end exponent, p(x) = 6 and q(x) = e-2x
      • While in fraction numerator d y over denominator d x end fraction plus fraction numerator y over denominator 2 x end fraction equals 12, p left parenthesis x right parenthesis equals fraction numerator 1 over denominator 2 x end fraction  and q(x) = 12
  • For an equation in standard form, the integrating factor is straight e to the power of   integral p open parentheses x close parentheses   straight d x end exponent

How do I use an integrating factor to solve a differential equation?

  • STEP 1: If necessary, rearrange the differential equation into standard form
  • STEP 2: Find the integrating factor
    • Note that you don’t need to include a constant of integration here when you integrate  ∫p(x) dx
  • STEP 3: Multiply both sides of the differential equation by the integrating factor
  • This will turn the equation into an exact differential equation of the form fraction numerator straight d over denominator straight d x end fraction open parentheses y straight e to the power of integral p open parentheses x close parentheses   straight d x end exponent close parentheses equals q open parentheses x close parentheses straight e to the power of   integral p open parentheses x close parentheses   straight d x end exponent
  • STEP 4: Integrate both sides of the equation with respect to x
    • The left side will automatically integrate to space y straight e to the power of integral p open parentheses x close parentheses   d x end exponent
    • For the right side, integrate integral q left parenthesis x right parenthesis straight e to the power of integral p left parenthesis x right parenthesis d x end exponent d x using your usual techniques for integration
    • Don’t forget to include a constant of integration
      • Although there are two integrals, you only need to include one constant of integration
  • STEP 5: Rearrange your solution to get it in the form y = f(x)

What else should I know about using an integrating factor to solve differential equations?

  • After finding the general solution using the steps above you may be asked to do other things with the solution
    • For example you may be asked to find the solution corresponding to certain initial or boundary conditions

Worked example

Consider the differential equation fraction numerator straight d y over denominator straight d x end fraction equals 2 x y plus 5 straight e to the power of x squared end exponent where  y = 7  when  x = 0.

Use an integrating factor to find the solution to the differential equation with the given boundary condition.

T8-Lavls_fm-8-1-2-integrating-factors-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.