Introduction to First-Order Ordinary Differential Equations

Applications

Phenomena in many disciplines are modeled by first-order ordinary differential
equations (odes). Some examples include

• Mechanical Systems
• Electrical Circuits
• Population Models
• Newton's Law of Cooling
• Compartmental Analysis
  

General Form

The general form of a first-order ordinary differential equation is

Here t is the independent variable and y(t) is the dependent variable. The goal is to
determine the unknown function y(t) whose derivative satisfies the above condition
and which passes through the point

Terminology

• ordinary differential equation: An ordinary differential equation (ode) is for
  which the unknown dependent variable is a function of one variable only.
  For example:

  

  is an ode. An example of a partial differential equation is

  

  Here the unknown dependent variable u(x,t) is a function of both x and t.

• order: The order of an ode is the order of the highest derivative in the
  equation. For example, the ode

  

  is a second-order ode.

• linear: A linear ode is one in which the dependent variable and its
  derivatives appear linearly. Example of a linear ode:

  

  This is a linear ode even though there are terms sin(t) and log(t). The
  independent variable t can appear nonlinearly in a linear ode. An ode that
  is not linear is nonlinear. Here is an example of a nonlinear ode:

  

  The general form of a linear ode is

  

• autonomous ode: For an autonomous ode f(t,y) is a function of y only.
  Here is an example of an autonomous ode:

  

Methods for Solving First-Order ODE

There are three principal methods for analyzing and solving differential
equations. These are

• Qualitative analysis
• Numerical approximation
• Analytic (exact) analysis
  

Most realistic odes cannot be solved exactly. For these problems one does
a qualitative analysis to get a rough idea of the behavior of the solution. Then
a numerical method is employed to get an accurate solution. In this way,
one can verify the answer obtained from the numerical method by comparing
it with the answer obtained from qualitative analysis. In a few fortunate
cases a first-order ode can be solved exactly.

[ODE Home] [1st-Order Home] [2nd-Order Home] [Laplace Transform Home] [Notation] [References]

Copyright © 1996 Department of Mathematics, Oregon State University

If you have questions or comments, don't hestitate to contact us.