Differential Equations with Boundary-Value Problems (MindTap Course List)
9th Edition
ISBN: 9781305965799
Author: Dennis G. Zill
Publisher: Cengage Learning
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Chapter 8.1, Problem 24E
To determine
To verify: The given vector
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9.
P =
15
-4
-7
2e31 – 8e-
-4e31 + 2e-
ž(1) = |
3e3t – 20e-
-6e31 + 5et
Show that x1 (t) is a solution to the system x = Px by evaluating derivatives and the
matrix product
-4
ž(1) = | 15 -7
Enter your answers in terms of the variable t.
Show that x2(t) is a solution to the system x' = Px by evaluating derivatives and the
matrix product
9.
3(1) = | 15
-4
X2(t)
-7
Enter your answers in terms of the variable t.
2. Given the following 2 x 2 linear system with constant coefficients
x' = Ax
(H)
x= Ax+g(t), (N)
where g is not the zero vector. Which of the following statements are true? Justify your
answers.
A. If , is a solution to (H) and 7, is a solution to (N), then , +27, is a solution to
(N).
B. If , and 2 are both solutions to (N), then ₁-2 is a solution to (H).
14. Assume x E R. Give the matrix associated with the quadratic form
10(x,) - 5x,x2 + 6(x,).
Chapter 8 Solutions
Differential Equations with Boundary-Value Problems (MindTap Course List)
Ch. 8.1 - In Problems 16 write the given linear system in...Ch. 8.1 - Prob. 2ECh. 8.1 - Prob. 3ECh. 8.1 - Prob. 4ECh. 8.1 - Prob. 5ECh. 8.1 - In Problems 16 write the given linear system in...Ch. 8.1 - In Problems 710 write the given linear system...Ch. 8.1 - Prob. 8ECh. 8.1 - In Problems 16 write the given linear system in...Ch. 8.1 - Prob. 10E
Ch. 8.1 - Prob. 11ECh. 8.1 - Prob. 12ECh. 8.1 - Prob. 13ECh. 8.1 - In Problems 1116 verify that the vector X is a...Ch. 8.1 - Prob. 15ECh. 8.1 - Prob. 16ECh. 8.1 - In Problems 1720 the given vectors are solutions...Ch. 8.1 - In Problems 1720 the given vectors are solutions...Ch. 8.1 - Prob. 19ECh. 8.1 - Prob. 20ECh. 8.1 - Prob. 21ECh. 8.1 - Prob. 22ECh. 8.1 - Prob. 23ECh. 8.1 - Prob. 24ECh. 8.1 - Prob. 25ECh. 8.1 - Prob. 26ECh. 8.2 - In Problems 112 find the general solution of the...Ch. 8.2 - In Problems 112 find the general solution of the...Ch. 8.2 - Prob. 3ECh. 8.2 - Prob. 4ECh. 8.2 - Prob. 5ECh. 8.2 - Prob. 6ECh. 8.2 - Prob. 7ECh. 8.2 - Prob. 8ECh. 8.2 - Prob. 9ECh. 8.2 - Distinct Real Eigenvalues In Problems 112 find the...Ch. 8.2 - Prob. 11ECh. 8.2 - Prob. 12ECh. 8.2 - Prob. 13ECh. 8.2 - Prob. 14ECh. 8.2 - In Problem 27 of Exercises 4.9 you were asked to...Ch. 8.2 - Prob. 19ECh. 8.2 - Prob. 20ECh. 8.2 - Prob. 21ECh. 8.2 - Prob. 22ECh. 8.2 - Prob. 23ECh. 8.2 - Prob. 24ECh. 8.2 - Prob. 25ECh. 8.2 - Prob. 26ECh. 8.2 - Prob. 27ECh. 8.2 - Prob. 28ECh. 8.2 - Prob. 29ECh. 8.2 - Prob. 30ECh. 8.2 - Prob. 31ECh. 8.2 - Prob. 32ECh. 8.2 - Prob. 33ECh. 8.2 - Prob. 34ECh. 8.2 - Prob. 35ECh. 8.2 - Prob. 36ECh. 8.2 - Prob. 37ECh. 8.2 - Prob. 38ECh. 8.2 - Prob. 39ECh. 8.2 - Prob. 40ECh. 8.2 - In Problems 3546 find the general solution of the...Ch. 8.2 - Prob. 42ECh. 8.2 - Prob. 43ECh. 8.2 - Prob. 44ECh. 8.2 - In Problems 3546 find the general solution of the...Ch. 8.2 - Prob. 46ECh. 8.2 - In Problems 47 and 48 solve the given...Ch. 8.2 - Prob. 48ECh. 8.2 - The system of mixing tanks shown in Figure 8.2.7...Ch. 8.2 - Prob. 50ECh. 8.2 - Prob. 51ECh. 8.2 - Prob. 53ECh. 8.2 - Show that the 5 5 matrix...Ch. 8.2 - Prob. 55ECh. 8.2 - Prob. 56ECh. 8.3 - In Problems 18 use the method of undetermined...Ch. 8.3 - Prob. 2ECh. 8.3 - Prob. 3ECh. 8.3 - Prob. 4ECh. 8.3 - In Problems 18 use the method of undetermined...Ch. 8.3 - Prob. 6ECh. 8.3 - Prob. 7ECh. 8.3 - Prob. 8ECh. 8.3 - Prob. 9ECh. 8.3 - Prob. 10ECh. 8.3 - Consider the large mixing tanks shown in Figure...Ch. 8.3 - Prob. 12ECh. 8.3 - Prob. 13ECh. 8.3 - Prob. 14ECh. 8.3 - Prob. 15ECh. 8.3 - Prob. 16ECh. 8.3 - Prob. 17ECh. 8.3 - Prob. 18ECh. 8.3 - Prob. 19ECh. 8.3 - Prob. 20ECh. 8.3 - Prob. 21ECh. 8.3 - Prob. 22ECh. 8.3 - Prob. 23ECh. 8.3 - Prob. 24ECh. 8.3 - Prob. 25ECh. 8.3 - Prob. 26ECh. 8.3 - Prob. 27ECh. 8.3 - Prob. 28ECh. 8.3 - Prob. 29ECh. 8.3 - Prob. 30ECh. 8.3 - Prob. 31ECh. 8.3 - Prob. 32ECh. 8.3 - Prob. 33ECh. 8.3 - Prob. 34ECh. 8.3 - The system of differential equations for the...Ch. 8.3 - Prob. 36ECh. 8.4 - Prob. 1ECh. 8.4 - Prob. 2ECh. 8.4 - Prob. 3ECh. 8.4 - Prob. 4ECh. 8.4 - In problem 58 use (1) use to find the general...Ch. 8.4 - In problem 58 use (1) use to find the general...Ch. 8.4 - Prob. 7ECh. 8.4 - Prob. 8ECh. 8.4 - Prob. 9ECh. 8.4 - Prob. 10ECh. 8.4 - Prob. 11ECh. 8.4 - Prob. 12ECh. 8.4 - Prob. 13ECh. 8.4 - Prob. 14ECh. 8.4 - Prob. 15ECh. 8.4 - Prob. 16ECh. 8.4 - In problem 1518 use the method of Example 2 to...Ch. 8.4 - Prob. 18ECh. 8.4 - Prob. 19ECh. 8.4 - Prob. 20ECh. 8.4 - Prob. 21ECh. 8.4 - Prob. 22ECh. 8.4 - Prob. 23ECh. 8.4 - Prob. 24ECh. 8.4 - Prob. 25ECh. 8.4 - Prob. 26ECh. 8 - fill in the blanks. 1. The vector X=k(45) is a...Ch. 8 - fill in the blanks. The vector...Ch. 8 - Consider the linear system X=(466132143)X. Without...Ch. 8 - Consider the linear system X = AX of two...Ch. 8 - In Problems 514 solve the given linear system. 5....Ch. 8 - Prob. 6RECh. 8 - Prob. 7RECh. 8 - Prob. 8RECh. 8 - Prob. 9RECh. 8 - Prob. 10RECh. 8 - Prob. 11RECh. 8 - Prob. 12RECh. 8 - Prob. 13RECh. 8 - Prob. 14RECh. 8 - Prob. 15RECh. 8 - Prob. 16RE
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- What can you conclude about the values of the quadratic form Q(x)?arrow_forward7. Consider the invertible matrix It is given that A-1 = b11 b21 b12 b22 2 -- (EE) b31 b32 1 2 -1 -- (7) = 2 -2 A 0 1 1 (a) Find the entry b21 of A-¹ using the adjoint formula. X (b) Solve the linear system AX + 2B = 0, where X = y B = " and 0 is the zero Z matrix of the appropriate size.arrow_forward5. Find the cofactors and the adjoint matrix of A where (1 1 -1 b) A =| 2 3 (7 0 -1 a) A = - 2 0 - 2 4 -3 5 4 5 6 2arrow_forward
- 2arrow_forwardThis is the first part of a four-part problem. Let P = 2e3t – 6e -4e3t + 2e 1(t) = [3et 2(t) = -6e3t + 5e] 15et a. Show that j1(t) is a solution to the system i' = Pỹ by evaluating derivatives and the matrix product 9 = 15 Enter your answers in terms of the variable t. b. Show that ğa(t) is a solution to the system j' = Pj by evaluating derivatives and the matrix product = Enter your answers in terms of the variable t. 8 ]- [8 ]arrow_forwardQ2. Find the basis and dimension of the solution space of the homogeneous linear system X1 – 4x2 + 3x3 – X4 = 0 2х, — 8х, + 6х; — 2х, —D 0 -2 2 3] Q3. Let, A =|-2 3 2. Find the eigenvalues and bases for the eigenspace of A and A-1. [-4 2 5arrow_forward
- Let A be 3x5 matrix. Then AX=b has always infinitly many solutions.arrow_forward6. Find the general solution of the system of differential equations -B -B X = dt -- -B х, -B -B where a 0 and B 0 are real nonzero constants. Hint: The characteristic polynomial of the coefficient matrix is -(A– a - B)²(A – a + 23).arrow_forwardThe matrix that projects onto the line y = -x is X 0.6 0.8 0.8 -0.6arrow_forward
- 2. In the linear system Ax = b, 27 b = 2 A = 6 (a) Show that x = be a solution of the linear system. 7 (b) Show that b can be expressed as a linear combination of columns of A with scalars 1, -1, and 7. (c) Find x A if it is defined. Whey x is not a solution of ATx = b?arrow_forward9. Find the solution vector of the given system by applying Gauss-Seidel method for 2 iterations with the intial vector x0) = (0,0,0)'. -2x1 + x2 +x3 = 4 X2 + 2x3 = 0 1 X1 – 2x2 -x3 = -4 A.) x(2) = (-1.0000, 1.0000, –0.5000) В) х(2) — (1.2500, —1.3333,0.6666)* C.) x2) = (-1.6250, –5.3750, 2.6875) D.) x(2) = (1.2500, –0.9166,0.0666)' E.) x(2) = (-1,6250, 1.3125, –0.6562)arrow_forward13 Solve the following linear system of DE; x' = Añ. 9x15x2 + 3x3 4x2 + 3x3 O 13arrow_forward
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