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 etc. Problems 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
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:
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
variable (variable in the lower half of all the derivative symbols)
variable - these are functions of the independent variable.
They appear in the upper half of the derivative expressions
In general differential
equations can be characterized as
In the following pages we will
look at first order, second order and a fourth order example
( problem parameters are a function of a single variable) or
Partial (problem parameters are a function of more than one
(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)
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)
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)
the DE (the highest derivative in the DE on the left)
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
Value Problem - all the BC'c are specified at the starting value
of the independent variable
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.
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.
linear ODEs are solved in these pages