Page 203 - Numerical Analysis Using MATLAB and Excel

Chapter 5 Differential Equations, State Variables, and State Equations

5.12 Summary

• Differential equations are classified by:
Type − Ordinary or Partial

Order − The highest order derivative which is included in the differential equation
Degree − The exponent of the highest power of the highest order derivative after the differen-
tial equation has been cleared of any fractions or radicals in the dependent variable and its
derivatives

• If the dependent variable is a function of only a single variable , that is, if y = f x() , the
x
y
differential equation which relates and is said to be an ordinary differential equation and
x
y
it is abbreviated as ODE.
(
,
y
• If the dependent variable is a function of two or more variables such as y = f x t ) , where x
and are independent variables, the differential equation that relates , , and is said to be
t
t
y x
a partial differential equation and it is abbreviated as PDE.
• A function y = f x() is a solution of a differential equation if the latter is satisfied when and
y
its derivatives are replaced throughout by fx() and its corresponding derivatives. Also, the
initial conditions must be satisfied.
•The ODE

n
m
d y d n – 1 y dy d x d m – 1 x dx
a --------- + a n – 1 ---------------- + … + a ------ + a y = b ---------- + b m – 1 ----------------- + … + b ------ + b x
m
0
1
0
n
1
dt n dt n – 1 dt dt m dt n – 1 dt
is a non−homogeneous differential equation if the right side, known as forcing function, is not
zero. If the forcing function is zero, the differential equation is referred to as homogeneous dif-
ferential equation.
•The most general solution of an homogeneous ODE is the linear combination
t
y () = k y t() + k y t() + k y t() + … + k y t()
n
n
3
3
2
1
H
1
2
where the subscript is used to denote homogeneous and k k k … k, 1 2 , 3 , , n are arbitrary con-
H
stants.
• Generally, in engineering the solution of the homogeneous ODE, also known as the comple-
mentary solution, is referred to as the natural response, and is denoted as y () or simply y N .
t
N
The particular solution of a non−homogeneous ODE is be referred to as the forced response,
and is denoted as y t() or simply y F . The total solution of the non−homogeneous ODE is the
F
summation of the natural and forces responses, that is,
5−42 Numerical Analysis Using MATLAB® and Excel®, Third Edition
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