Introduction to Second-Order ODE

Many physical phenomena are modeled by second order ODE's. Some
examples are:

General Form

The general form for a second order ordinary differential equation is

on some interval [a,b]. Here, t is the independent variable and y(t) is the
dependent variable. The goal is to find functions that satisfy the above
ordinary differential equation.

Usually, the ode is accompanied by initial conditions or boundary conditions.
Initial conditions have the form:

Here t_0 is some starting time and y_0 and y'_0 are values of the the function
and its derivative at t_0. Alternatively, boundary conditions can be specified.
These involve specifying the function and or its derivative at two different
values of t. For example,

Methods for Solving Second-Order ODE

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

Most second-order odes arising in realistic applications 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 to the answer obtained from
qualitative analysis. In a few fortunate cases a second-order ode can be
solved exactly.

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

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