# Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

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,» = (J). e = ( j).

Find two linearly independent solutions x(1)(t) and x(2)(t) of Eq. (i).

(b) To find a third solution assume that x = gte1 + qel; then show that g and q must

satisfy

(A - I)g = 0, (iii)

(A - I)q = g. (iv)

(c) Show that g = Cjg(1) + c2g(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 Cj = c2.

(d) It is convenient to choose Cj = c2 = 2. For this choice show that

g=(-2) (-?) <v)

where we have dropped the multiples of g(1) and g(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 g(1) in the first column and with the eigenvector

g 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 ^ J, where X is an arbitrary real number.

(a) Find J2, J3, and J4.

" X" "X"-

(b) Use an inductive argument to show that J" = I _ . n

0X

(c) Determine exp (Jt).

(d) Use exp(Jt) to solve the initial value problem x' = Jx, x(0) = x0.

20. Let

X00 J = I 0 X 1 I ,

\0 0 Xj

where X is an arbitrary real number.

(a) Find J2, J3, and J4.

(b) Use an inductive argument to show that

(X" 0 0 \

J" = I 0 X" "X"1 I .

\0 0 X" J

(c) Determine exp (Jt).

(d) Observe that if you choose X = 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(Jt) with X = 1. Is the resulting matrix the same as the fundamental matrix ^(t) in Problem 18(e)? If not, explain the discrepancy.

21. Let

J

X1 0X t0 0

0I

X

where X is an arbitrary real number.

(a) Find J2, J3, and J4.

(b) Use an inductive argument to show that

J"

X" "X"-1 ["(" --- 1)/2]X"

0 X" "X"-1

0 0 X"

"-2

(c) Determine exp (Jt).

(d) Observe that if you choose X = 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(Jt) with X = 2. Observe that the resulting matrix is the same as the fundamental matrix 'ΤΥΞ in Problem 17(e).

7.9 Nonhomogeneous Linear Systems

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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 ΰ < t < β. By

the same argument as in Section 3.6 (see also Problem 16 in this section) the general

solution ofEq. (1) can be expressed as

x = clx(1)(t) + ... + cn x(n)(t) + v(t), (2)

where cxx(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

x = Ty. (4)

Then substituting for x in Eq. (3), we obtain

Ty' = ATy + g(t).

By multiplying by T -1 it follows that

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