Particular Solutions (AQA A Level Maths: Pure): Revision Note

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

What is a boundary condition and an 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

• 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

• It is possible to have two boundary conditions

  • e.g. a particle that starts from rest at the origin has velocity v=0 and displacement x=0, at time t=0

    • "Initially at rest" only tells you that v=0 at t=0, it does not necessarily mean the acceleration is zero

  • For a second order differential equation you need two boundary conditions to find the particular solution

    • Integrating twice gives two constants of integration

    • So two conditions are needed to find them both