AQA A Level Maths: Pure复习笔记8.3.2 Particular Solutions

Particular Solutions

What is a particular solution?

  • Ensure you are familiar with General Solutions first
  • With extra information, the constant of integration, c, can be found
  • This means the particular solution (from the family of solutions) can be found

8.3.2-Notes-ps_eg

What is a boundary condition/initial condition?

  • A boundary condition is a piece of extra information that lets you find the particular solution
    • For example knowing y = 4 when x = 0 in the preceding example
    • In a model this could be a particle coming to rest after a certain time, ie v = 0 at time t

8.3.2-Notes-ps_bound_cond

  • Differential equations are used in modelling, experiments and real-life situations
  • A boundary condition is often called an initial condition when it gives the situation at the start of the model or experiment
    • This is often linked to time, so t = 0

8.3.2-Notes-ps_init_cond

  • It is possible to have two boundary conditions
    • eg a particle initially at rest has velocity, v = 0 and acceleration, a = 0 at time, t = 0
    • for a second order differential equation you need two boundary conditions to find the particular solution

Worked Example

8.3.2-Example-soltn_a8.3.2-Example-soltn_b

 

 

 

 

 

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