Explanation Given: The linear system is X ′ = ( − 1 1 − 2 1 ) X + ( 1 cot t ) . Calculation: The system is in the form X ′ = A X + F ( t ) . Write the characteristic equation of the coefficient matrix A . ( A − λ I ) = ( − 1 − λ 1 − 2 1 − λ ) For the Eigen values, determinant of the coefficient matrix equate to zero. Det ( A − λ I ) = | − 1 − λ 1 − 2 1 − λ | ( − 1 − λ ) ( 1 − λ ) + 2 = 0 − ( 1 − λ 2 ) + 2 = 0 λ 2 − 1 + 2 = 0 The values of λ is calculated as follows: λ 2 + 1 = 0 λ = ± i λ 1 = i or λ 2 = − i where λ 1 , λ 2 are the two Eigen values. For λ 1 = i , the Eigen vector is given below. ( A − i I | 0 ) = ( − 1 − i 1 0 − 2 1 − i 0 ) Therefore, the value is ( − 1 − i ) K 1 + K 2 = 0 From the above expression K 2 = ( 1 + i ) K 1 . The Corresponding Eigen vector is K 1 = ( 1 1 + i ) . Consider A is the coefficient matrix having real values of homogeneous system and K is an Eigen vector corresponding the complex Eigen value λ 1 = i then the solution is X = K e λ 1 t and this solution will come in the form of X = X 1 + X 2 . X = ( 1 1 + i ) e i t = ( 1 1 + i ) ( cos t + i sin t ) = ( cos t + i sin t cos t + i sin t + i cos t − sin t ) = ( cos t cos t − sin t ) + i ( sin t sin t + cos t ) The solution vector of the system is given below. X 1 = ( cos t cos t − sin t ) , X 2 = ( sin t sin t + cos t ) Therefore, the complementary solution is X c = C 1 ( cos t cos t − sin t ) + C 2 ( sin t sin t + cos t ) The solution vectors can be written as follows: ϕ ( t ) = ( cos t sin t cos t − sin t sin t + cos t ) Det ϕ ( t ) = ( cos t ) ( sin t + cos t ) − ( sin t ) ( cos t − sin t ) = cos t sin t + cos 2 t − ( sin t cos t − sin 2 t ) Determination of ϕ − 1 ( t ) is calculated as follows: ϕ − 1 ( t ) = ( sin t + cos t − sin t sin t − cos t cos t ) Express the particular solution X p as given below. X p = ϕ ( t ) ∫ ϕ − 1 ( t ) F ( t ) d t

A First Course in Differential Equations with Modeling Applications (MindTap Course List)

11th Edition

ISBN: 9781305965720

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 8, Problem 13RE

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The solution of the linear system X′=(−11−21)X+(1cott).

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Problem 2. Consider the equation: x?y"(x) – xy' +y = 0. Given that yı(x) = x is a solution of this equation. Use the method of reduction of order, find the second solution y2(x) of the equation so that y1 and y2 are linearly independent. (Hint: y2(x) should be given in the form y2(x) = u(x)y1(x). Substitute it into the equation to find u(x).) %3D

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The solution of the linear system X ′ = ( − 1 1 − 2 1 ) X + ( 1 cot t ) .

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The solution of the linear system X′=(−11−21)X+(1cott).

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