Edexcel A Level Further Maths: Core Pure:复习笔记8.1.2 Solving First Order Differential Equations

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

Worked Example

Consider the differential equation  where  y = 7  when  x = 0.

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