Edexcel A Level Further Maths: Core Pure:复习笔记8.1.1 Intro to Differential Equations

General Solutions

What is a differential equation?

• Any equation, involving a derivative term, is a differential equation
• Equations involving only first derivative terms are called first order differential equations
• Equations involving second derivative terms are called second order differential equations

What is a general solution?

• Integration will be involved in solving the differential equation
• ie working back to “y = f(x)”

• A constant of integration, c is produced
• This gives an infinite number of solutions to the differential equation, each of the form y = g(x) + c  (ie  y = f(x)  where  f(x) = g(x) + c)
• … and the solution y= g(x) + c is called the general solutionThese are often called a family of solutions …

Worked Example

Find the general solution to the differential equation .

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/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
• 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

The velocity of a particle, initially at rest, is modelled by the differential equation   , where is the velocity of the particle and  is the time since the particle began moving.

a) Find the velocity of the particle after 3 seconds.

b) Find the time at which the particle comes to rest for the second time.