Introduction to Second-Order Ordinary Differential Equations

Applications

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

• Mechanical Systems/Vibrations (springs, pendulums, etc)
• Electrical Circuits
• One Dimensional Motion
• Pursuit Problems

General Form

The general form for a second order ordinary differential equation is

on some interval [a,b]. Here, x is the independent variable and y(x) 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

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

Most second-order odes arising in realist applications cannot be solved exactly. For these problems one does a qualitative analysis to get a rough ide 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 the answer obtained from qualitative analysis. In a few fortunate cases a second-order ode can be solved exactly.

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