Explanation Given: The given non-homogeneous system is, X ′ = ( 1 − 1 1 3 ) X + ( t t + 1 ) and X ( 0 ) = ( 3 2 ) Calculation: For the given system, the coefficient matrix is A = ( 1 − 1 1 3 ) and F ( t ) = ( t t + 1 ) . Since, the characteristic equation for the matrix A is | ( A − λ I ) | = 0 where I is the Identity matrix. Substitute the values of matrix A and I in above expression. | 1 − λ − 1 1 3 − λ | = 0 ( 1 − λ ) ( 3 − λ ) + 1 = 0 λ 2 − 4 λ + 4 = 0 ( λ − 2 ) 2 = 0 Therefore, the Eigen value of the above system is λ 1 = 2 with multiplicity 2. Now, to obtain the Eigen vector K 1 corresponding to the Eigen value λ 1 = 2 . Solve the equation ( A − λ 1 I ) K 1 = 0 where, K 1 = ( x 1 y 1 ) . Substituting the values of A , λ ` 1 and K 1 in the above equation gives, [ ( 1 − 1 1 3 ) − ( 2 0 0 2 ) ] ⋅ ( x 1 y 1 ) = ( 0 0 ) ( − 1 − 1 1 1 ) ⋅ ( x 1 y 1 ) = ( 0 0 ) Applying the row transformation, R 1 → R 1 + R 2 in above system, ( 0 0 1 1 ) ⋅ ( x 1 y 1 ) = ( 0 0 ) ( 0 x 1 + y 1 ) = ( 0 0 ) Solving above equations gives, x 1 = − y 1 Assume that y 1 = n for some scalar n which implies x 1 = − n . Therefore, the value of ( x 1 y 1 ) is, ( x 1 y 1 ) = ( − n n ) = n ( − 1 1 ) Therefore the eigenvector K 1 corresponding to the Eigen value λ 1 = 2 is ( − 1 1 ) . Now, to obtain the second Eigen vector K 2 . Solve the equation ( A − λ 2 I ) K 2 = K 1 where K 2 = ( x 2 y 2 ) . Substituting the values of A , λ ` 2 , K 2 and K 1 in the above equation gives, [ ( 1 − 1 1 3 ) − ( 2 0 0 2 ) ] ⋅ ( x 2 y 2 ) = ( − 1 1 ) ( − 1 − 1 1 1 ) ⋅ ( x 2 y 2 ) = ( − 1 1 ) Applying the row transformation, R 1 → R 1 + R 2 in above system, ( 0 0 1 1 ) ⋅ ( x 2 y 2 ) = ( − 1 1 ) ( 0 x 2 + y 2 ) = ( 0 1 ) Solving above equations gives, x 2 = 1 − y 2 Assume that y 2 = m for some scalar m which implies x 2 = 1 − m . Therefore, the value of ( x 2 y 2 ) is, ( x 2 y 2 ) = ( 1 − m m ) = m ( − 1 1 ) + ( 1 0 ) Since, from the above calculation, the eigenvector K 1 obtained is ( − 1 1 ) . Therefore the second eigenvector K 2 is ( 1 0 ) . The complementary function for the given non homogeneous system is, X c = c 1 K 1 e λ 1 t + c 2 [ K 1 t + K 2 ] e λ 1 t . Substitute the value of λ 1 , K 1 and K 2 in above equation, X c = c 1 ( − 1 1 ) e 2 t + c 2 [ ( − 1 1 ) t + ( 1 0 ) ] e 2 t Since the function F ( t ) is given as F ( t ) = ( t t + 1 ) = ( 1 1 ) t + ( 0 1 ) Therefore, to obtain the particular solution X p for the given system, assume that X p is given of the form, X p = ( a 2 b 2 ) t + ( a 1 b 1 ) , where a 1 , b 1 are constants

Differential Equations with Boundary-Value Problems

9th Edition

ISBN: 9781337632515

Author: Dennis G. Zill

Publisher: Cengage Learning US

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Chapter 8.3, Problem 10E

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The table below shows the acreage, number of visitors, and total revenue of state parks and recreational areas in Massachusetts, New York, and Vermont in 2010. State Acreage (in thousands) Visitors (in thousands) Revenue (in thousands) Massachusetts 350 35,271 $12,644 New York 1,354 56,322 $85,558 Vermont 69 758 $10,969 Select the three true statements based on the data in the table. A. Vermont had the highest revenue per acre of state parks and recreational areas. B. Vermont had approximately 11 visitors per acre of state parks and recreational areas. C. New York had the highest number of visitors per acre of state parks and recreational areas. D. Massachusetts had approximately 36 visitors per acre of state parks and recreational areas. E. New York had revenue of approximately $63.19 per acre of state parks and recreational areas. F. Massachusetts had revenue of approximately $0.03 per acre of state parks and recreational areas.

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Chapter 8 Solutions

Differential Equations with Boundary-Value Problems

Ch. 8.1 - In Problems 16 write the given linear system in...Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 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. 8E Ch. 8.1 - In Problems 16 write the given linear system in...Ch. 8.1 - Prob. 10E

Ch. 8.1 - Prob. 11E Ch. 8.1 - Prob. 12E Ch. 8.1 - Prob. 13E Ch. 8.1 - In Problems 1116 verify that the vector X is a...Ch. 8.1 - Prob. 15E Ch. 8.1 - Prob. 16E Ch. 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. 19E Ch. 8.1 - Prob. 20E Ch. 8.1 - Prob. 21E Ch. 8.1 - Prob. 22E Ch. 8.1 - Prob. 23E Ch. 8.1 - Prob. 24E Ch. 8.1 - Prob. 25E Ch. 8.1 - Prob. 26E Ch. 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. 3E Ch. 8.2 - Prob. 4E Ch. 8.2 - Prob. 5E Ch. 8.2 - Prob. 6E Ch. 8.2 - Prob. 7E Ch. 8.2 - Prob. 8E Ch. 8.2 - Prob. 9E Ch. 8.2 - Distinct Real Eigenvalues In Problems 112 find the...Ch. 8.2 - Prob. 11E Ch. 8.2 - Prob. 12E Ch. 8.2 - Prob. 13E Ch. 8.2 - Prob. 14E Ch. 8.2 - In Problem 27 of Exercises 4.9 you were asked to...Ch. 8.2 - Prob. 19E Ch. 8.2 - Prob. 20E Ch. 8.2 - Prob. 21E Ch. 8.2 - Prob. 22E Ch. 8.2 - Prob. 23E Ch. 8.2 - Prob. 24E Ch. 8.2 - Prob. 25E Ch. 8.2 - Prob. 26E Ch. 8.2 - Prob. 27E Ch. 8.2 - Prob. 28E Ch. 8.2 - Prob. 29E Ch. 8.2 - Prob. 30E Ch. 8.2 - Prob. 31E Ch. 8.2 - Prob. 32E Ch. 8.2 - Prob. 33E Ch. 8.2 - Prob. 34E Ch. 8.2 - Prob. 35E Ch. 8.2 - Prob. 36E Ch. 8.2 - Prob. 37E Ch. 8.2 - Prob. 38E Ch. 8.2 - Prob. 39E Ch. 8.2 - Prob. 40E Ch. 8.2 - In Problems 3546 find the general solution of the...Ch. 8.2 - Prob. 42E Ch. 8.2 - Prob. 43E Ch. 8.2 - Prob. 44E Ch. 8.2 - In Problems 3546 find the general solution of the...Ch. 8.2 - Prob. 46E Ch. 8.2 - In Problems 47 and 48 solve the given...Ch. 8.2 - Prob. 48E Ch. 8.2 - The system of mixing tanks shown in Figure 8.2.7...Ch. 8.2 - Prob. 50E Ch. 8.2 - Prob. 51E Ch. 8.2 - Prob. 53E Ch. 8.2 - Show that the 5 5 matrix...Ch. 8.2 - Prob. 55E Ch. 8.2 - Prob. 56E Ch. 8.3 - In Problems 18 use the method of undetermined...Ch. 8.3 - Prob. 2E Ch. 8.3 - Prob. 3E Ch. 8.3 - Prob. 4E Ch. 8.3 - In Problems 18 use the method of undetermined...Ch. 8.3 - Prob. 6E Ch. 8.3 - Prob. 7E Ch. 8.3 - Prob. 8E Ch. 8.3 - Prob. 9E Ch. 8.3 - Prob. 10E Ch. 8.3 - Consider the large mixing tanks shown in Figure...Ch. 8.3 - Prob. 12E Ch. 8.3 - Prob. 13E Ch. 8.3 - Prob. 14E Ch. 8.3 - Prob. 15E Ch. 8.3 - Prob. 16E Ch. 8.3 - Prob. 17E Ch. 8.3 - Prob. 18E Ch. 8.3 - Prob. 19E Ch. 8.3 - Prob. 20E Ch. 8.3 - Prob. 21E Ch. 8.3 - Prob. 22E Ch. 8.3 - Prob. 23E Ch. 8.3 - Prob. 24E Ch. 8.3 - Prob. 25E Ch. 8.3 - Prob. 26E Ch. 8.3 - Prob. 27E Ch. 8.3 - Prob. 28E Ch. 8.3 - Prob. 29E Ch. 8.3 - Prob. 30E Ch. 8.3 - Prob. 31E Ch. 8.3 - Prob. 32E Ch. 8.3 - Prob. 33E Ch. 8.3 - Prob. 34E Ch. 8.3 - The system of differential equations for the...Ch. 8.3 - Prob. 36E Ch. 8.4 - Prob. 1E Ch. 8.4 - Prob. 2E Ch. 8.4 - Prob. 3E Ch. 8.4 - Prob. 4E Ch. 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. 7E Ch. 8.4 - Prob. 8E Ch. 8.4 - Prob. 9E Ch. 8.4 - Prob. 10E Ch. 8.4 - Prob. 11E Ch. 8.4 - Prob. 12E Ch. 8.4 - Prob. 13E Ch. 8.4 - Prob. 14E Ch. 8.4 - Prob. 15E Ch. 8.4 - Prob. 16E Ch. 8.4 - In problem 1518 use the method of Example 2 to...Ch. 8.4 - Prob. 18E Ch. 8.4 - Prob. 19E Ch. 8.4 - Prob. 20E Ch. 8.4 - Prob. 21E Ch. 8.4 - Prob. 22E Ch. 8.4 - Prob. 23E Ch. 8.4 - Prob. 24E Ch. 8.4 - Prob. 25E Ch. 8.4 - Prob. 26E Ch. 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. 6RE Ch. 8 - Prob. 7RE Ch. 8 - Prob. 8RE Ch. 8 - Prob. 9RE Ch. 8 - Prob. 10RE Ch. 8 - Prob. 11RE Ch. 8 - Prob. 12RE Ch. 8 - Prob. 13RE Ch. 8 - Prob. 14RE Ch. 8 - Prob. 15RE Ch. 8 - Prob. 16RE

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Solution Summary: The author calculates the coefficient matrix for the given non-homogeneous system. Substitute the values of matrix A and I in above expression.

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Chapter 8 Solutions