Download annot_002_25.pdf

Survey
yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

page
1
page
2
page
3
page
4
page
5
page
6
page
7
page
8
page
9
page
10
page
11
page
12
Document related concepts
no text concepts found
Transcript 
Section 5.1
Copyright©Arunabha Biswas
265
Section 5.1
Copyright©Arunabha Biswas
266
Ex: Find a particular solution of the previous problem that
satisfies the initial conditions: x1 (0) = 10, x2 (0) = 12 and
x3 (0) = −1.
Section 5.1
Copyright©Arunabha Biswas
267
Section 5.1
Copyright©Arunabha Biswas
268
Section 5.1
Copyright©Arunabha Biswas
269
Section 5.2: The Eigen Value Method for Homogeneous System
Section 5.2
Copyright©Arunabha Biswas
270
Now just consider the homogeneous system of equations
dx
= Ax or x0 = Ax where A is a constant matrix. Here we
dt
can use the ‘auxiliary equation’ type technic to find the form of
the general solution.
Section 5.2
Copyright©Arunabha Biswas
271
Definition (Eigen Values and Eigen Vectors)
The number λ is called an eigenvalue of the n × n matrix A
provided that
|A − λI|= 0.
An eigenvector associated with the eigenvalue λ is a nonzero
vector v such that Av = λv, so that
(A − λI)v = 0.
The equation |A − λI|= 0 (which is a polynomial) is known as
the characteristic polynomial of the matrix A.
Section 5.2
Copyright©Arunabha Biswas
272
Theorem (Eigenvalue Solution of x0 = Ax)
Let λ be an eigenvalue of the constant coefficient matrix A of
dx
the first order linear system
= Ax or x0 = Ax. If v is an
dt
eigenvector associated with λ, then
x(t) = veλt
is a nontrivial solution of the system.
Section 5.2
Copyright©Arunabha Biswas
273
Eigenvalue Method with Real Eigenvalues:
Given the first order liner system x0 = Ax where A is a n × n
constant matrix
1. Solve the characteristic polynomial |A − λI|= 0 for
eigenvalues λ1 , λ2 , · · · λn of A and assume all of them are
real numbers.
2. Find the associated eigenvectors (linearly independent) v1 ,
v2 , · · ·, vn .
3. In this case the general solution of x0 = Ax is
x(t) = c1 v1 eλ1 t + c2 v2 eλ2 t + · · · cn vn eλn t
where the xk = ck vk eλk t , k = 1, 2, · · · n are the linearly
independent solutions.
Section 5.2
Copyright©Arunabha Biswas
274
Ex: Solve the system:
x01 = 2x1 + 3x2 , x02 = 2x1 + x2 .
Section 5.2
Copyright©Arunabha Biswas
275
Section 5.2
Copyright©Arunabha Biswas
276
Related documents