# Elementary differential equations 7th edition - Boyce W.E

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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