Page 167 - Numerical Analysis Using MATLAB and Excel

Chapter 5 Differential Equations, State Variables, and State Equations

5.3 Solutions of Ordinary Differential Equations (ODE)

A function y = f x() is a solution of a differential equation if the latter is satisfied when and its
y
derivatives are replaced throughout by fx() and its corresponding derivatives. Also, the initial
conditions must be satisfied.
For example a solution of the differential equation

2
d y y = 0
-------- +
dx 2
is

x
y = k sin + k cos x
2
1
since and its second derivative satisfy the given differential equation.
y
Any linear, time-invariant system can be described by an ODE which has the form

–
n
d y d n1 y dy
a --------- + a n – 1 ---------------- + … + a ------ + a y
0
1
n
–
dt n dt n1 dt
m m – 1
d x d x dx
b ---------- + b ----------------- + … + b ------ + b x (5.12)
= m dt m m – 1 dt n – 1 1 dt 0
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
Excitation Forcing Function x t()
(
)
NON HOMOGENEOUS DIFFERENTIAL EQUATION
–
If the excitation in (B12) is not zero, that is, if xt() ≠ 0 , the ODE is called a non-homogeneous
ODE. If xt() = 0 , it reduces to:
n
d y d n – 1 y dy
a --------- + a n – 1 ---------------- + … + a ------ + a y = 0
n
0
1
dt n dt n – 1 dt (5.13)
HOMOGENEOUS DIFFERENTIAL EQUATION
The differential equation of (5.13) above is called a homogeneous ODE and has different linearly
n
independent solutions denoted as y t() y t() y t() … y t(),,, 2 3 , n .
1
We will now prove that the most general solution of (5.13) is:
y () = k y t() + k y t() + k y t() + … + k y t() (5.14)
t
H
1
3
3
2
n
n
1
2
where the subscript on the left side is used to emphasize that this is the form of the solution of
H
the homogeneous ODE and k k k … k, 1 2 , 3 , , n are arbitrary constants.
5−6 Numerical Analysis Using MATLAB® and Excel®, Third Edition
Copyright © Orchard Publications