ODE: Overview

Ordinary Differential Equation (ODE) are among the most important mathematical quantity used in engineering analysis.  They are used across the discipline for design and simulation. The term "ordinary" is used for mathematical quantities that are a function of a single variable - velocity is a function of time, pressure is a function of depth etcProblems in dynamics, vibrations, particle trajectory are naturally described as ODE. .  You must have already come across many ODE's from different courses so far.  In ODE's you will encounter an equation withderivative(s)  called the differential equation (DE). Along with the differential equations (DE) you will also require boundary conditions (BC).  Without BC the problem specification is incomplete.   Typically, an ODE is used to express the mathematical model of the engineering problem that can capture the changes in the system or its properties. There are two variables in an ODE:

1. Independent variable (variable in the lower half of all the derivative symbols)
2. Dependent variable - these are functions of the independent variable.  They appear in the upper half of the derivative expressions
The DE is usually written with all the dependent variable terms on the left of the equal sign.  The solution of the ODE is to determine the dependent variables as functions of the independent variable between a start value and an end value

In general differential equations can be characterized as

• Ordinary ( problem parameters are a function of a single variable) or Partial (problem parameters are a function of more than one variable)
• Linear (the terms in the in the DE are linear - dependent variables and their derivatives are not multiplied with each other and all of the derivatives are raised to the order of 1 or 0) and Nonlinear (when even a single term in the equation is not linear)
• Homogenous (the DE has a value of 0 right of the equal sign) and Non homogenous (the right hand side is a function of the independent variable or its power)
• Constant Coefficient (the terms on the left in the DE are multiplied by constant values) Variable coefficient ( the terms may include functions of the independent variable)
• Order of the DE (the highest derivative in the DE on the left)
In the following pages we will look at first order, second order and a fourth order example

Boundary Conditions

Boundary conditions (BC) are essential to ODE.  The number of boundary conditions required is the same as the order of the DE.  There are two types of ODE's with regard to the BC

• Initial Value Problem - all the BC'c are specified at the starting value of the independent variable
• Boundary Value Problem - some BC's are specified at the initial point and some at other point.  If the other point is the final point then it is a two-point boundary variable problem.  Otherwise it is a multipoint value problem.
Solution

If the DE is linear, the solution to a homogenous DE is called the natural motion, the solution to the non-homogenous DE is called the particular solution.  The total solution is the sum of the two solutions.  The homogenous solution includes undetermined constants.  The BC's are applied to the total solution to identify these constants.

Only linear ODEs are solved in these pages