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 an equation 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.
  That is no powers (besides 1), no products, and no nonlinear functions of the unknown
  dependent variable or any of its derivatives appear in the equation.
  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 first-order 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 integration
• 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.

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