Explanation Given: The nonhomogeneous linear system is X ′ = ( − 1 − 1 − 1 1 ) X + ( 1 1 ) t 2 + ( 4 − 6 ) t + ( − 1 5 ) . The general solution of the above nonhomogeneous linear system is X = c 1 ( 1 − 1 − 2 ) e 2 t + c 2 ( 1 − 1 + 2 ) e − 2 t + ( 1 0 ) t 2 + ( − 2 4 ) t + ( 1 0 ) . Calculation: Consider the homogeneous system associated with nonhomogeneous linear system in X ′ = A X form, where A = ( − 1 − 1 − 1 1 ) . The general solution of the nonhomogeneous system is X = X c + X p . Therefore, the given general solution can be written as follows: X c = c 1 ( 1 − 1 − 2 ) e 2 t + c 2 ( 1 − 1 + 2 ) e − 2 t X p = ( 1 0 ) t 2 + ( − 2 4 ) t + ( 1 0 ) Check X c = c 1 ( 1 − 1 − 2 ) e 2 t + c 2 ( 1 − 1 + 2 ) e − 2 t is the solution of the X ′ = A X . From the given complementary solution of the homogenous system, assume two solution vectors are as follows: X 1 = ( 1 − 1 − 2 ) e 2 t X 2 = ( 1 − 1 + 2 ) e − 2 t Differentiate each vector solution as follows: X ′ 1 = ( 1 − 1 − 2 ) ( 2 e 2 t ) X ′ 2 = ( 1 − 1 + 2 ) ( − 2 e − 2 t ) Simplify, each differentiate vector as follows: X ′ 1 = ( 2 e 2 t − 2 e 2 t − 2 e 2 t ) X ′ 2 = ( − 2 e − 2 t 2 e − 2 t − 2 e − 2 t ) Calculate the value of A X 1 as follows: A X 1 = ( − 1 − 1 − 1 1 ) ( 1 − 1 − 2 ) e 2 t = ( 2 − 2 − 2 ) e 2 t = ( 2 e 2 t − 2 e 2 t − 2 e 2 t ) = X ′ 1 From above calculation, it is proved that X 1 is the solution of homogenous system X ′ = A X . Calculate the value of A X 2 as follows: A X 2 = ( − 1 − 1 − 1 1 ) ( 1 − 1 + 2 ) e − 2 t = ( − 2 − 2 + 2 ) e − 2 t = ( − 2 e − 2 t 2 e − 2 t − 2 e − 2 t ) = X ′ 2 From above calculation, it is proved that X 2 is the solution of homogenous system X ′ = A X . So, each of X 1 , X 2 satisfy the given homogenous system. Now, calculate the wronskian W ( X 1 , X 2 ) as follows: W ( X 1 , X 2 ) = | e 2 t e − 2 t ( − 1 − 2 ) e 2 t ( − 1 + 2 ) e − 2 t | = ( − 1 + 2 ) − ( − 1 − 2 ) = − 1 + 2 + 1 + 2 = 2 2 The vectors are linearly independent, since wronskian W ( X 1 , X 2 ) ≠ 0

WEB ASSIGN FOR ZILL'S DIFFERENTIAL EQUAT

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ISBN: 9780357539545

Author: ZILL

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Chapter 8.1, Problem 26E

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To prove: The general solution of the nonhomogeneous linear system X′=(−1−1−11)X+(11)t2+(4−6)t+(−15) on the interval (−∞,∞) is X=c1(1−1−2)e2t+c2(1−1+2)e−2t+(10)t2+(−24)t+(10).

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Chapter 8.1, Problem 25E

Chapter 8.2, Problem 1E

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Chapter 8 Solutions

WEB ASSIGN FOR ZILL'S DIFFERENTIAL EQUAT

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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To prove: The general solution of the nonhomogeneous linear system X ′ = ( − 1 − 1 − 1 1 ) X + ( 1 1 ) t 2 + ( 4 − 6 ) t + ( − 1 5 ) on the interval ( − ∞ , ∞ ) is X = c 1 ( 1 − 1 − 2 ) e 2 t + c 2 ( 1 − 1 + 2 ) e − 2 t + ( 1 0 ) t 2 + ( − 2 4 ) t + ( 1 0 ) .

Solution Summary: The author explains the general solution of the nonhomogeneous linear system.

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To prove: The general solution of the nonhomogeneous linear system X′=(−1−1−11)X+(11)t2+(4−6)t+(−15) on the interval (−∞,∞) is X=c1(1−1−2)e2t+c2(1−1+2)e−2t+(10)t2+(−24)t+(10).

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Chapter 8 Solutions