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 provides an

opportunity to apply math to engineering applications such as heat transfer and vibrations.

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 terms:

What is the order of the differential equation ?

Is the differential equation an Ordinary or Partial differential equation ?

Is the differential equation. Linear or Nonlinear ?

Does the differential equation have Constant or Variable Coefficients ?

Is the differential equation Homogeneous or Non-homogeneous?

Is the differential equation Separable or non-separable?

Is the differential equation exact?

Let’s answer these questions (detailed 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. (do to the term dy/dx )

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

A differential equation is classified as ordinary if it contains only one independent

variable, x. The two examples in the table above are both ordinary differential equations.

A differential equation that has more than one independent variable, say x and t, is a

partial differential equation.

i.e. The differential equation given below is a partial differential equation involving

the two independent variables, x and t.

∂u/∂t = k ∂²u/∂x² here u(x,t) is the dependent variable. It might represent the

temperature distribution in a rod as a function of position, x, along the rod and the

time, t; k is a constant

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