Books
in black and white
Main menu
Share a book About us Home
Books
Biology Business Chemistry Computers Culture Economics Fiction Games Guide History Management Mathematical Medicine Mental Fitnes Physics Psychology Scince Sport Technics
Ads

Elementary differential equations 7th edition - Boyce W.E

Boyce W.E Elementary differential equations 7th edition - Wiley publishing , 2001. - 1310 p.
ISBN 0-471-31999-6
Download (direct link): elementarydifferentialequat2001.pdf
Previous << 1 .. 170 171 172 173 174 175 < 176 > 177 178 179 180 181 182 .. 486 >> Next

17. Consider the system
(a) Show that r = 2 is an eigenvalue of multiplicity 3 of the coefficient matrix A and that there is only one corresponding eigenvector, namely,
15. Show that all solutions of the system
V C RC>
(a) Show that the eigenvalues are real and equal if L = 4 R2 C.
(i)
7.8 Repeated Eigenvalues
409
(b) Using the information in part (a), write down one solution x(1)(t) of the system (i). There is no other solution of the purely exponential form x = ?ert.
(c) To find a second solution assume that x = ?te2t + qe2t. Show that ? and q satisfy the equations
(A - 2I)? = 0, (A - 2l)q = ?
Since ? has already been found in part (a), solve the second equation for q. Neglect the multiple of ?(1) that appears in q, since it leads only to a multiple of the first solution x(1). Then write down a second solution x(2) (t) of the system (i).
(d) To find a third solution assume that x = ?(t2/2)e2t + qte2t + ?e2t. Show that ?, q, and ? satisfy the equations
(A - 2I)? = 0, (A - 2l)q = ?, (A - 2I)? = q-
The first two equations are the same as in part (c), so solve the third equation for ?, again
neglecting the multiple of ?(1) that appears. Then write down a third solution x(3) (t) of the system (i).
(e) Write down a fundamental matrix ^(f) for the system (i).
(f) Form a matrix T with the eigenvector ?(1 ) in the first column, and the generalized
eigenvectors q and ? in the second and third columns. Then find T-1 and form the product J = T-1 AT. The matrix J is the Jordan form of A.
18. Consider the system
( 5 -3 -2)
x' = Ax = I 8 -5 -4 I x- (i)
V-4 3 I)
(a) Show that r = 1 is a triple eigenvalue of the coefficient matrix A, and that there are only two linearly independent eigenvectors, which we may take as
,=(j). e=(
Find two linearly independent solutions x(1) (t) and x(2) (t) of Eq. (i).
(b) To find a third solution assume that x = ?tel + qel; then show that ? and q must
satisfy
(A - I)? = 0, (iii)
(A - I)q = ? (iv)
(c) Show that ? = Cj?(1) + c2?(2), where Cj and c2 are arbitrary constants, is the most general solution of Eq. (iii). Show that in order to solve Eq. (iv) it is necessary that cy = c2.
(d) It is convenient to choose cy = c2 = 2. For this choice show that
? = (-2) (-?) <v)
where we have dropped the multiples of ?(1) and ?(2) that appear in q. Use the results given in Eqs. (v) to find a third linearly independent solution x(3) (t) of Eq. (i).
(e) Write down a fundamental matrix ^(f) for the system (i).
(f) Form a matrix T with the eigenvector ?(1) in the first column and with the eigenvector ? and the generalized eigenvector q from Eqs. (v) in the other two columns. Find T-1 and
form the product J = T-1 AT. The matrix J is the Jordan form of A.
410
Chapter 7. Systems of First Order Linear Equations
19. Let J = ^0 1J, where k is an arbitrary real number.
(a) Find J2, J3, and J4.
(kn
(b) Use an inductive argument to show that Jn = I _ . n
\ U k
(c) Determine exp (Jf).
(d) Use exp(Jf) to solve the initial value problem x' = Jx, x(0) = xU.
20. Let
(k U U\
J = I U k 1 I ,
\0 U k)
where k is an arbitrary real number.
(a) Find J2, J3, and J4.
(b) Use an inductive argument to show that
(kn U 0 \
Jn = I U kn nkn-1 I .
\U U kn J
(c) Determine exp (Jf).
(d) Observe that if you choose k = 1, then the matrix J in this problem is the same as the matrix J in Problem 18(f). Using the matrix T from Problem 18(f), form the product T exp(Jf) with k = 1. Is the resulting matrix the same as the fundamental matrix ^(f) in Problem 18(e)? If not, explain the discrepancy.
21. Let
where k is an arbitrary real number.
(a) Find J2, J3, and J4.
(b) Use an inductive argument to show that
(kn nkn-1 [n(n - 1)/2]kn-2'
Jn = I 0 kn nkn-1
\0 0 kn
(c) Determine exp (Jf).
(d) Observe that if you choose k = 2, then the matrix J in this problem is the same as the matrix J in Problem 17(f). Using the matrix T from Problem 17(f), form the product T exp(Jf) with k = 2. Observe that the resulting matrix is the same as the fundamental matrix ^(f) in Problem 17(e).
7.9 Nonhomogeneous Linear Systems
411
7.9 Nonhomogeneous Linear Systems
In this section we turn to the nonhomogeneous system
x = P(t)x + g(t), (1)
where the n x n matrix P(t) and n x 1 vector g(t) are continuous for a < t < ft. By
the same argument as in Section 3.6 (see also Problem 16 in this section) the general
solution of Eq. (1) can be expressed as
x = c1x(1)(t) +-----+ cn x(n) (t) + v(t), (2)
where c1x(1)(t) + ? + cnx(n)(t) is the general solution of the homogeneous system
x; = P(t)x, and v(t) is a particular solution of the nonhomogeneous system (1). We will briefly describe several methods for determining v(t).
Diagonalization. We begin with systems of the form
X = Ax + g(t), (3)
where A is an n x n diagonalizable constant matrix. By diagonalizing the coefficient matrix A, as indicated in Section 7.7, we can transform Eq. (3) into a system of equations that is readily solvable.
Let T be the matrix whose columns are the eigenvectors ?(1),..., ?(n) of A, and define a new dependent variable y by
Previous << 1 .. 170 171 172 173 174 175 < 176 > 177 178 179 180 181 182 .. 486 >> Next