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# 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
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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
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