Ordinary Differential Equations

An ODE is used to express the rate of change of one quantity with respect to another. You have probably been working with ODEs since you began your scientific or engineering course work. One defining characteristic of an ODE is that its derivatives are a function of one independent variable. A general form of a first-order ODE is shown in Eq. (20.1).

Equation 20.1

The order of a differential is defined as the order of the highest derivative appearing in the equation. Ordinary differential equations can be of any order. A general form of a second-order ODE is shown in Eq. (20.2).

Equation 20.2

Any higher-order ODE can be expressed as a coupled set of first-order differential equations. For example, the second-order ODE shown in Eq. (20.2) can be reduced to a coupled set of two first-order differential equations.

Equation 20.3

The second expression in Eq. (20.3) looks trivial in that the left-hand side is the same as the right-hand side, but the ODE solvers we will discuss later in this chapter use the coupled first-order form of the ODE in their solution process. The ODE solvers would integrate the first-order equations shown in Eq. (20.3) to obtain values for the dependent variables y and dy / dx as a function of the independent variable x .