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

Boyce W.E. Elementary Differential Equations and Boundary Value Problems - John Wiley & Sons, 2001. - 1310 p.
Download (direct link): elementarydifferentialequations2001.pdf
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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
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 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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