MathHelp

Introduction - to Differential Equations (d.e.’s)

In a Nut Shell: A differential equation is an equation where at least one term in the

equation involves a derivative. Differential equations describe mechanical, electrical,

and fluid systems in engineering. The subject of differential equations gives you an

opportunity to apply math to engineering problems. Enjoy.

The method of solution for a differential equation (d.e.) depends on its type. So

you need to be able to classify the type of differential equation. Classifications

include the following:

What is the order of the d.e.?

Is the d.e. an Ordinary or Partial d.e?

Is the d.e. Linear or Nonlinear?

Does the d.e. have Constant or Variable Coefficients?

Is the d.e. Homogeneous or Non-homogeneous?

Is the d.e. Separable or non-separable?

Let’s answer these questions (below).

The order of a d.e. is the order of the highest derivative of the dependent variable

in the d.e. In the examples below y is the dependent variable and x is the

independent variable. i.e.

dy/dx = 5 is a first order d.e. (due to the term dy/dx )

d²y/dx² + dy/dx + y = 3x is a second order d.e. (due to the term d²y/dx² )

A d.e. is classified as ordinary if it contains only one independent variable, x.

The two examples in section 3 above are both ordinary d.e.’s.

A d.e. that has more than one independent variable is a partial d.e. i.e. The d .e.

given below is a partial d.e. involving two independent variables, x and t.

∂u/∂t = k ∂²u/∂x² ; here u(x,t) is the temperature, dependent variable, in a rod

as a function of position, x, and time, t; k is a constant

Return to Differential Equations Notes