Solving First Order Differential Equations (Edexcel A Level Further Maths): Revision Note

First Order Differential Equations

What is a differential equation?

• A differential equation is simply an equation that contains derivatives

  • For example  is a differential equation

  • And so is 

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  is a first order differential equation

  • But  is not a first order differential equation, because it contains the second derivative 

• 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  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 

  • Be careful – the ‘functions of x’ p(x) and q(x) may just be constants!

    • For example in , p(x) = 6 and q(x) = e^-2x

    • While in ,   and q(x) = 12

• For an equation in standard form, the integrating factor is 

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 

• STEP 4: Integrate both sides of the equation with respect to x

  • The left side will automatically integrate to 

  • For the right side, integrate  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