AQA A Level Maths: Pure复习笔记8.3.3 Separation of Variables

Separation of Variables

What does separation of variables mean?


  • Many differential equations used in modelling either …
    • … have two variables involved (ie x and y), or,
    • ... involve a function of the dependent variable (ie y) only
  • This is particularly true where proportionality is involved
    • eg population change is dependent on both time and the size of the population


  • This type of question is covered in more detail in Modelling with Differential Equations

How do I know if I need to separate the variable in a question?


  • There is a product of functions in different variables
    • ie dy/dx = f(x) × g(y)
  • It will not be possible to integrate directly from an equation in the form dy/dx= g(y)

How do I solve a separating variables question?


  • STEP 1: Separate all y terms on one side and all x terms on the other side
  • STEP 2: Integrate both sides
  • STEP 3: Include one “overall” constant of integration
  • STEP 4: Use the initial or boundary condition to find the particular solution
  • STEP 5: Write the particular solution in sensible, or required, format

Worked Example