5.2 Classification

Differential equations are classified by:

1. Type - Ordinary or Partial

2. Order - The highest order derivative which is included in the differential equation

3. Degree - The exponent of the highest power of the highest order derivative after the differential equation has been cleared of any fractions or radicals in the dependent variable and its derivatives

For example, the differential equation

is an ordinary differential equation of order 4 and degree 2.

If the dependent variable y is a function of only a single variable x, that is, if y = f( x), the differential equation which relates y and x is said to be an ordinary differential equation and it is abbreviated as ODE.

The differential equation

is an ODE with constant coefficients.

The differential equation

is an ODE with variable coefficients.

If the dependent variable y is a function of two or more variables such as y = f( x, t), where x and t are independent variables, the differential equation that relates y, x, and t is said to be a partial differential equation and it is abbreviated as PDE.

An example of a partial differential equation is the well-known one-dimensional wave equation shown below.

Most engineering problems are solved with ordinary differential equations with constant coefficients; however, partial differential equations provide often quick solutions to some practical applications as illustrated with...