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 …
Find the general solution to the differential equation .
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
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.