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?…

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