Differential Equations

Differential Equations

Definition

  • A differential equation is a mathematical equation that relates some function with its derivatives.
  • It involves concepts of calculus to describe the problem or phenomenon.
  • These equations can be categorised as ordinary or partial, depending on whether they contain ordinary derivatives or partial derivatives.

Types of Differential Equations

  • First order differential equations: Equations involving the first derivative of a function but no higher derivatives.
  • Second order differential equations: Involves second derivatives.
  • Homogeneous differential equations: These are equations which are structurally the same as their corresponding homogeneous equation.

Formulating a Differential Equation

  • To create a differential equation, you need to have a physical or mathematical problem which involves rates of change, or similar concepts, that can be modelled effectively.
  • The specifics of the problem will direct the differential terms present in the equation.
  • The solution of the problem will dictate the structure of the equation.

Prescribed Solution to a Differential Equation

  • These usually involve finding the general solution or particular solution.
  • A general solution contains constants which can represent any real number value.
  • Particular solutions involve specified values for the problem, which can be found by substitute given constants into the general solution.
  • Although differential equations may seem daunting, they can be solved using known techniques from integration and differentiation.

Separation of Variables

  • A common method to solve first order differential equations.
  • This technique involves algebraic manipulations to separate and isolate the variables on different sides of the equation for easy integration.

Applications of Differential Equations

  • Differential equations play a critical role in fields like physics, engineering, economics, and biology where they help model and predict natural phenomena.
  • In physics, they are often used to describe phenomena like motion and waves.
  • In economics, they can model economic growth and decay, or help predict the behaviour of markets.
  • In biology, they can be used to model population dynamics and the spread of diseases.