Explanation Consider the matrix, [ 1 − 1 − 1 1 ] ... ( 1 ) The characteristic Equation of matrix A is. | 1 − r − 1 − 1 1 − r | = 0 ( 1 − r ) 2 − 1 = 0 r = 0 , 2 The Eigen vector corresponding to r = 0 are, u 1 = [ 1 1 ] x 1 ( t ) = [ 1 1 ] The Eigen vector corresponding to r = 2 are, u 2 = [ 1 − 1 ] x 2 ( t ) = [ e 2 t − e 2 t ] The solution of the system is, x h ( t ) = c 1 x 1 ( t ) + c 2 x 2 ( t ) Substitute [ 1 1 ] for x 1 ( t ) and [ e 2 t − e 2 t ] for x 2 ( t ) in Equation (2). x h ( t ) = c 1 [ 1 1 ] + c 2 [ e 2 t − e 2 t ] The matrix form system is, x ′ ( t ) = A x ( t ) + [ − 3 1 ] ... ( 3 ) The form of the particular solution is, x p ( t ) = t a + b … ( 4 ) Substitute t a + b for x p ( t ) in Equation (3). a = [ 1 − 1 − 1 1 ] ( t a + b ) + [ − 3 1 ] … ( 5 ) Substitute [ a 1 a 2 ] for a and [ b 1 b 2 ] for b in Equation (5). [ a 1 a 2 ] = [ 1 − 1 − 1 1 ] ( t [ a 1 a 2 ] + [ b 1 b 2 ] ) + [ − 3 1 ] [ a 1 a 2 ] = t [ a 1 − a 2 − a 1 + a 2 ] + [ b 1 − b 2 − 3 − b 1 + b 2 + 1 ] Equating the coefficient of like terms.

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Chapter 9.7, Problem 28E

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The particular solution set for the system x′(t)=[1−1−11]x(t)+[−31].

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Chapter 9.7, Problem 27E

Chapter 9.7, Problem 29E

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The particular solution set for the system x ′ ( t ) = [ 1 − 1 − 1 1 ] x ( t ) + [ − 3 1 ] .

Solution Summary: The author explains that the particular solution set for the system is xprime (t)= left[cc1& -1

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The particular solution set for the system x′(t)=[1−1−11]x(t)+[−31].

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