Explanation Given: The given non-homogeneous system is, X ′ = ( 0 1 − 1 0 ) X + ( 0 sec t tan t ) Calculation: The system is given in the form X ′ = A X + F ( t ) (1). Consider, the characteristic equation of a matrix A is solved using | A − λ I | = 0 , where I is identity matrix. Now, the characteristic equation of the coefficient matrix A is | ( A − λ I ) | = 0 | − λ 1 − 1 − λ | = 0 λ 2 + 1 = 0 λ = ± i So, the Eigen values are λ 1 = i , λ 2 = − i , . Now evaluate the Eigen vector for λ 1 = 1 + i as, ( A − ( i ) I 0 ) = ( − i 1 0 − 1 − i 0 ) On solving it gives, − i K 1 + K 2 = 0 On further evaluation it becomes i K 1 = K 2 . Thus, the corresponding Eigen vector is K = ( 1 i ) . Consider the coefficient matrix A having real values of homogeneous system and K be an Eigen vector corresponding to the complex Eigen value λ 1 , then the solution is X c = K e λ 1 t Thus, the value of complementary function X c is X c = ( 1 i ) e i t = ( 1 i ) ( cos t + i sin t ) = ( cos t − sin t ) + i ( sin t cos t ) On comparing with equation X c = X 1 + i X 2 , it gives the solution vectors of given system as X 1 = ( cos t − sin t ) X 2 = ( sin t cos t ) Now, the entries of X 1 form the first column of solution matrix ϕ ( t ) and the entries of X 2 form the second column of ϕ ( t ) , which means the value of ϕ ( t ) is ϕ ( t ) = ( cos t sin t − sin t cos t ) Now, evaluating determinant of the above matrix ϕ ( t ) gives det [ ϕ ( t ) ] = | cos t sin t − sin t cos t | = cos 2 t + sin 2 t = 1 Now the value of ϕ − 1 ( t ) is, ϕ − 1 ( t ) = 1 | ϕ ( t ) | ⋅ ( Adj . ϕ ( t ) ) ϕ − 1 ( t ) = 1 | ϕ ( t ) | ⋅ [ Adj

Student Solutions Manual For Zill's A First Course In Differential Equations With Modeling Applications, 11th

11th Edition

ISBN: 9781305965737

Author: Dennis G. Zill

Publisher: Brooks Cole

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Chapter 8.3, Problem 27E

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To solve: The given non-homogeneous system by using variation of parameters.

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Chapter 8.3, Problem 26E

Chapter 8.3, Problem 28E

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Student Solutions Manual For Zill's A First Course In Differential Equations With Modeling Applications, 11th

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The given non-homogeneous system by using variation of parameters.

Solution Summary: The author calculates the non-homogeneous system by using variation of parameters. The characteristic equation of a matrix A is solved using left|A-lambda Iright|=0

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To solve: The given non-homogeneous system by using variation of parameters.

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