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nagle_differential_equations_ISM_Part58

# nagle_differential_equations_ISM_Part58 - Exercises 9.4 6...

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Exercises 9.4 6. x 1 ( t ) x 2 ( t ) = 0 1 - 1 0 x 1 ( t ) x 2 ( t ) + 0 t 2 8. x 1 ( t ) x 2 ( t ) x 3 ( t ) = 0 1 0 0 0 1 - 1 1 0 x 1 ( t ) x 2 ( t ) x 3 ( t ) + 0 0 cos t 10. x 1 ( t ) = 2 x 1 ( t ) + x 2 ( t ) + te t ; x 2 ( t ) = - x 1 ( t ) + 3 x 2 ( t ) + e t 12. x 1 ( t ) = x 2 ( t ) + t + 3; x 2 ( t ) = x 3 ( t ) - t + 1; x 3 ( t ) = - x 1 ( t ) + x 2 ( t ) + 2 x 3 ( t ) + 2 t 14. Linearly independent 16. Linearly dependent 18. Linearly independent 20. Yes. 3 e - t e 4 t 2 e - t - e 4 t ; c 1 3 e - t 2 e - t + c 2 e 4 t - e 4 t 22. Linearly independent; fundamental matrix is e t sin t - cos t e t cos t sin t e t - sin t cos t The general solution is c 1 e t e t e t + c 2 sin t cos t - sin t + c 3 - cos t sin t cos t 24. c 1 e 3 t 0 e 3 t + c 2 - e 3 t e 3 t 0 + c 3 - e - 3 t - e - 3 t e - 3 t + 5 t + 1 2 t 4 t + 2 281

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Chapter 9 28. X - 1 ( t ) = 1 2 e t - 1 2 e t 1 2 e - 5 t 1 2 e - 5 t ; x ( t ) = 2 e - t + e 5 t - 2 e - t + e 5 t 32. Choosing x 0 = col(1 , 0 , 0 , . . . , 0), then x 0 = col(0 , 1 , 0 , . . . , 0), and so on, the correspond- ing solutions x 1 , x 2 , . . . , x n will have a nonvanishing Wronskian at the initial point t 0 . Hence, { x 1 , x 2 , . . . , x n } is a fundamental solution set EXERCISES 9.5: Homogeneous Linear Systems with Constant Coefficients 2. Eigenvalues are r 1 = 3 and r
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nagle_differential_equations_ISM_Part58 - Exercises 9.4 6...

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