Section 8.1

1. $Φ (t) = [\begin{array}{r} e^{t} & e^{3 t} \\ - e^{t} & e^{3 t} \end{array}], x (t) = \frac{1}{2} [\begin{array}{r} 5 e^{t} + e^{3 t} \\ - 5 e^{t} + e^{3 t} \end{array}]$ $Φ (t) = [\begin{array}{r} e^{t} & e^{3 t} \\ - e^{t} & e^{3 t} \end{array}], x (t) = \frac{1}{2} [\begin{array}{r} 5 e^{t} + e^{3 t} \\ - 5 e^{t} + e^{3 t} \end{array}]$
2. $Φ (t) = [\begin{array}{r} 1 & e^{4 t} \\ 2 & - 2 e^{4 t} \end{array}], x (t) = \frac{1}{4} [\begin{array}{r} 3 + 5 e^{4 t} \\ 6 - 10 e^{4 t} \end{array}]$ $Φ (t) = [\begin{array}{r} 1 & e^{4 t} \\ 2 & - 2 e^{4 t} \end{array}], x (t) = \frac{1}{4} [\begin{array}{r} 3 + 5 e^{4 t} \\ 6 - 10 e^{4 t} \end{array}]$
3. $Φ (t) = [\begin{array}{r} 5 \cos 4 t & - 5 \sin 4 t \\ 2 \cos 4 t + 4 \sin 4 t & 4 \cos 4 t - 2 \sin 4 t \end{array}], x (t) = \frac{1}{4} [\begin{array}{r} - 5 \sin 4 t \\ 4 \cos 4 t - 2 \sin 4 t \end{array}]$ $Φ (t) = [\begin{array}{r} 5 \cos 4 t & - 5 \sin 4 t \\ 2 \cos 4 t + 4 \sin 4 t & 4 \cos 4 t - 2 \sin 4 t \end{array}], x (t) = \frac{1}{4} [\begin{array}{r} - 5 \sin 4 t \\ 4 \cos 4 t - 2 \sin 4 t \end{array}]$
4. $Φ (t) = e^{2 t} [\begin{array}{c} 1 & 1 + t \\ 1 & t \end{array}], x (t) = e^{2 t} [\begin{array}{c} 1 + t \\ t \end{array}]$ $Φ (t) = e^{2 t} [\begin{array}{c} 1 & 1 + t \\ 1 & t \end{array}], x (t) = e^{2 t} [\begin{array}{c} 1 + t \\ t \end{array}]$
5. $Φ (t) = [\begin{array}{r} 2 \cos 3 t & - 2 \sin 3 t \\ - 3 \cos 3 t + 3 \sin 3 t & 3 \cos 3 t + 3 \sin 3 t \end{array}], x (t) = \frac{1}{3} [\begin{array}{r} 3 \cos 3 t - \sin 3 t \\ - 3 \cos 3 t + 6 \sin 3 t \end{array}]$ $Φ (t) = [\begin{array}{r} 2 \cos 3 t & - 2 \sin 3 t \\ - 3 \cos 3 t + 3 \sin 3 t & 3 \cos 3 t + 3 \sin 3 t \end{array}], x (t) = \frac{1}{3} [\begin{array}{r} 3 \cos 3 t - \sin 3 t \\ - 3 \cos 3 t + 6 \sin 3 t \end{array}]$
6. $Φ (t) = e^{5 t} [\begin{array}{r} \cos 4 t - 2 \sin 4 t & 2 \cos 4 t + 2 \sin 4 t \\ 2 \cos 4 t & 2 \sin 4 t \end{array}], x (t) = 2 e^{5 t} [\begin{array}{r} \cos 4 t + \sin 4 t \\ \sin 4 t \end{array}]$ $Φ (t) = e^{5 t} [\begin{array}{r} \cos 4 t - 2 \sin 4 t & 2 \cos 4 t + 2 \sin 4 t \\ 2 \cos 4 t & 2 \sin 4 t \end{array}], x (t) = 2 e^{5 t} [\begin{array}{r} \cos 4 t + \sin 4 t \\ \sin 4 t \end{array}]$
7. $Φ (t) = [\begin{array}{r} 6 & 3 e^{t} & 2 e^{- t} \\ 2 & e^{t} & e^{- t} \\ 5 & 2 e^{t} & 2 e^{- t} \end{array}], x (t) = [\begin{array}{r} - 12 + 12 e^{t} + 2 e^{- t} \\ - 4 + 4 e^{t} + e^{- t} \\ - 10 + 8 e^{t} + 2 e^{- t} \end{array}]$ $Φ (t) = [\begin{array}{r} 6 & 3 e^{t} & 2 e^{- t} \\ 2 & e^{t} & e^{- t} \\ 5 & 2 e^{t} & 2 e^{- t} \end{array}], x (t) = [\begin{array}{r} - 12 + 12 e^{t} + 2 e^{- t} \\ - 4 + 4 e^{t} + e^{- t} \\ - 10 + 8 e^{t} + 2 e^{- t} \end{array}]$
8. $Φ (t) = [\begin{array}{c} 0 & e^{t} & e^{3 t} \\ e^{- 2 t} & - e^{t} & - e^{3 t} \\ - e^{- 2 t} & 0 & e^{3 t} \end{array}], x (t) = [\begin{array}{c} e^{t} \\ - e^{t} + e^{- 2 t} \\ - e^{- 2 t} \end{array}]$ $Φ (t) = [\begin{array}{c} 0 & e^{t} & e^{3 t} \\ e^{- 2 t} & - e^{t} & - e^{3 t} \\ - e^{- 2 t} & 0 & e^{3 t} \end{array}], x (t) = [\begin{array}{c} e^{t} \\ - e^{t} + e^{- 2 t} \\ - e^{- 2 t} \end{array}]$
9. $e^{A t} = [\begin{array}{r} 2 e^{3 t} - e^{t} & - 2 e^{3 t} + 2 e^{t} \\ e^{3 t} - e^{t} & - e^{3 t} + 2 e^{t} \end{array}]$ $e^{A t} = [\begin{array}{r} 2 e^{3 t} - e^{t} & - 2 e^{3 t} + 2 e^{t} \\ e^{3 t} - e^{t} & - e^{3 t} + 2 e^{t} \end{array}]$
10. $e^{A t} = [\begin{array}{r} - 2 + 3 e^{2 t} & 3 - 3 e^{2 t} \\ - 2 + 2 e^{2 t} & 3 - 2 e^{2 t} \end{array}]$ $e^{A t} = [\begin{array}{r} - 2 + 3 e^{2 t} & 3 - 3 e^{2 t} \\ - 2 + 2 e^{2 t} & 3 - 2 e^{2 t} \end{array}]$
11. $e^{A t} = [\begin{array}{r} 3 e^{3 t} - 2 e^{2 t} & - 3 e^{3 t} + 3 e^{2 t} \\ 2 e^{3 t} - 2 e^{2 t} & - 2 e^{3 t} + 3 e^{2 t} \end{array}]$ $e^{A t} = [\begin{array}{r} 3 e^{3 t} - 2 e^{2 t} & - 3 e^{3 t} + 3 e^{2 t} \\ 2 e^{3 t} - 2 e^{2 t} & - 2 e^{3 t} + 3 e^{2 t} \end{array}]$
12. $e^{A t} = [\begin{array}{r} - 3 e^{t} + 4 e^{2 t} & 4 e^{t} - 4 e^{2 t} \\ - 3 e^{t} + 3 e^{2 t} & 4 e^{t} - 3 e^{2 t} \end{array}]$ $e^{A t} = [\begin{array}{r} - 3 e^{t} + 4 e^{2 t} & 4 e^{t} - 4 e^{2 t} \\ - 3 e^{t} + 3 e^{2 t} & 4 e^{t} - 3 e^{2 t} \end{array}]$
13. $e^{A t} = [\begin{array}{r} 4 e^{3 t} - 3 e^{t} & - 4 e^{3 t} + 4 e^{t} \\ 3 e^{3 t} - 3 e^{t} & - 3 e^{3 t} + 4 e^{t} \end{array}]$ $e^{A t} = [\begin{array}{r} 4 e^{3 t} - 3 e^{t} & - 4 e^{3 t} + 4 e^{t} \\ 3 e^{3 t} - 3 e^{t} & - 3 e^{3 t} + 4 e^{t} \end{array}]$
14. $e^{A t} = [\begin{array}{c} - 8 e^{t} + 9 e^{2 t} & 6 e^{t} - 6 e^{2 t} \\ - 12 e^{t} + 12 e^{2 t} & 9 e^{t} - 8 e^{2 t} \end{array}]$ $e^{A t} = [\begin{array}{c} - 8 e^{t} + 9 e^{2 t} & 6 e^{t} - 6 e^{2 t} \\ - 12 e^{t} + 12 e^{2 t} & 9 e^{t} - 8 e^{2 t} \end{array}]$
15. $e^{A t} = [\begin{array}{r} 5 e^{2 t} - 4 e^{t} & - 10 e^{2 t} + 10 e^{t} \\ 2 e^{2 t} - 2 e^{t} & - 4 e^{2 t} + 5 e^{t} \end{array}]$ $e^{A t} = [\begin{array}{r} 5 e^{2 t} - 4 e^{t} & - 10 e^{2 t} + 10 e^{t} \\ 2 e^{2 t} - 2 e^{t} & - 4 e^{2 t} + 5 e^{t} \end{array}]$
16. $e^{A t} = [\begin{array}{r} - 9 e^{t} + 10 e^{2 t} & 15 e^{t} - 15 e^{2 t} \\ - 6 e^{t} + 6 e^{2 t} & 10 e^{t} - 9 e^{2 t} \end{array}]$ $e^{A t} = [\begin{array}{r} - 9 e^{t} + 10 e^{2 t} & 15 e^{t} - 15 e^{2 t} \\ - 6 e^{t} + 6 e^{2 t} & 10 e^{t} - 9 e^{2 t} \end{array}]$
17. $e^{A t} = \frac{1}{2} [\begin{array}{r} e^{4 t} + e^{2 t} & e^{4 t} - e^{2 t} \\ e^{4 t} - e^{2 t} & e^{4 t} + e^{2 t} \end{array}]$ $e^{A t} = \frac{1}{2} [\begin{array}{r} e^{4 t} + e^{2 t} & e^{4 t} - e^{2 t} \\ e^{4 t} - e^{2 t} & e^{4 t} + e^{2 t} \end{array}]$
18. $e^{A t} = \frac{1}{2} [\begin{array}{r} e^{2 t} + e^{6 t} & - e^{2 t} + e^{6 t} \\ - e^{2 t} + e^{6 t} & e^{2 t} + e^{6 t} \end{array}]$ $e^{A t} = \frac{1}{2} [\begin{array}{r} e^{2 t} + e^{6 t} & - e^{2 t} + e^{6 t} \\ - e^{2 t} + e^{6 t} & e^{2 t} + e^{6 t} \end{array}]$
19. $e^{A t} = \frac{1}{5} [\begin{array}{r} 4 e^{10 t} + e^{5 t} & 2 e^{10 t} - 2 e^{5 t} \\ 2 e^{10 t} - 2 e^{5 t} & e^{10 t} + 4 e^{5 t} \end{array}]$ $e^{A t} = \frac{1}{5} [\begin{array}{r} 4 e^{10 t} + e^{5 t} & 2 e^{10 t} - 2 e^{5 t} \\ 2 e^{10 t} - 2 e^{5 t} & e^{10 t} + 4 e^{5 t} \end{array}]$
20. $e^{A t} = \frac{1}{5} [\begin{array}{r} e^{5 t} + 4 e^{15 t} & - 2 e^{5 t} + 2 e^{15 t} \\ - 2 e^{5 t} + 2 e^{15 t} & 4 e^{5 t} + e^{15 t} \end{array}]$ $e^{A t} = \frac{1}{5} [\begin{array}{r} e^{5 t} + 4 e^{15 t} & - 2 e^{5 t} + 2 e^{15 t} \\ - 2 e^{5 t} + 2 e^{15 t} & 4 e^{5 t} + e^{15 t} \end{array}]$
21. $e^{A t} = [\begin{array}{r} 1 + t & - t \\ t & 1 - t \end{array}]$ $e^{A t} = [\begin{array}{r} 1 + t & - t \\ t & 1 - t \end{array}]$
22. $e^{A t} = [\begin{array}{r} 1 + 6 t & 4 t \\ - 9 t & 1 - 6 t \end{array}]$ $e^{A t} = [\begin{array}{r} 1 + 6 t & 4 t \\ - 9 t & 1 - 6 t \end{array}]$
23. $e^{A t} = [\begin{array}{r} 1 + t & - t & - t - t^{2} \\ t & 1 - t & t - t^{2} \\ 0 & 0 & 1 \end{array}]$ $e^{A t} = [\begin{array}{r} 1 + t & - t & - t - t^{2} \\ t & 1 - t & t - t^{2} \\ 0 & 0 & 1 \end{array}]$
24. $e^{A t} = [\begin{array}{r} 1 + 3 t & 0 & - 3 t \\ 5 t + 18 t^{2} & 1 & 7 t - 18 t^{2} \\ 3 t & 0 & 1 - 3 t \end{array}]$ $e^{A t} = [\begin{array}{r} 1 + 3 t & 0 & - 3 t \\ 5 t + 18 t^{2} & 1 & 7 t - 18 t^{2} \\ 3 t & 0 & 1 - 3 t \end{array}]$
25. $e^{A t} = [\begin{array}{r} e^{2 t} & 5 t e^{2 t} \\ 0 & e^{2 t} \end{array}],$ $e^{A t} = [\begin{array}{r} e^{2 t} & 5 t e^{2 t} \\ 0 & e^{2 t} \end{array}],$ $x (t) = e^{A t} [\begin{array}{r} 4 \\ 7 \end{array}]$ $x (t) = e^{A t} [\begin{array}{r} 4 \\ 7 \end{array}]$
26. $e^{A t} = [\begin{array}{r} e^{7 t} & 0 \\ 11 t e^{7 t} & e^{7 t} \end{array}], x (t) = e^{A t} [\begin{array}{r} 5 \\ - 10 \end{array}]$ $e^{A t} = [\begin{array}{r} e^{7 t} & 0 \\ 11 t e^{7 t} & e^{7 t} \end{array}], x (t) = e^{A t} [\begin{array}{r} 5 \\ - 10 \end{array}]$
27. $e^{A t} = [\begin{array}{r} e^{t} & 2 t e^{t} & (3 t + 2 t^{2}) e^{t} \\ 0 & e^{t} & 2 t e^{t} \\ 0 & 0 & e^{t} \end{array}], x (t) = e^{A t} [\begin{array}{r} 4 \\ 5 \\ 6 \end{array}]$ $e^{A t} = [\begin{array}{r} e^{t} & 2 t e^{t} & (3 t + 2 t^{2}) e^{t} \\ 0 & e^{t} & 2 t e^{t} \\ 0 & 0 & e^{t} \end{array}], x (t) = e^{A t} [\begin{array}{r} 4 \\ 5 \\ 6 \end{array}]$
28. $e^{A t} = [\begin{array}{c} e^{5 t} & 0 & 0 \\ 10 t e^{5 t} & e^{5 t} & 0 \\ (20 t + 150 t) e^{5 t} & 30 t e^{5 t} & e^{5 t} \end{array}], x (t) = e^{A t} [\begin{array}{r} 40 \\ 50 \\ 60 \end{array}]$ $e^{A t} = [\begin{array}{c} e^{5 t} & 0 & 0 \\ 10 t e^{5 t} & e^{5 t} & 0 \\ (20 t + 150 t) e^{5 t} & 30 t e^{5 t} & e^{5 t} \end{array}], x (t) = e^{A t} [\begin{array}{r} 40 \\ 50 \\ 60 \end{array}]$
29. $e^{A t} = [\begin{array}{r} 1 & 2 t & 3 t + 6 t^{2} & 4 t + 6 t^{2} + 4 t^{3} \\ 0 & 1 & 6 t & 3 t + 6 t^{2} \\ 0 & 0 & 1 & 2 t \\ 0 & 0 & 0 & 1 \end{array}] e^{t}, x (t) = e^{A t} [\begin{array}{r} 1 \\ 1 \\ 1 \\ 1 \end{array}]$ $e^{A t} = [\begin{array}{r} 1 & 2 t & 3 t + 6 t^{2} & 4 t + 6 t^{2} + 4 t^{3} \\ 0 & 1 & 6 t & 3 t + 6 t^{2} \\ 0 & 0 & 1 & 2 t \\ 0 & 0 & 0 & 1 \end{array}] e^{t}, x (t) = e^{A t} [\begin{array}{r} 1 \\ 1 \\ 1 \\ 1 \end{array}]$
30. $e^{A t} = e^{3 t} [\begin{array}{r} 1 & 0 & 0 & 0 \\ 6 t & 1 & 0 & 0 \\ 9 t + 18 t^{2} & 6 t & 1 & 0 \\ 12 t + 54 t^{2} + 36 t^{3} & 9 t + 18 t^{2} & 6 t & 1 \end{array}], x (t) = e^{A t} [\begin{array}{r} 1 \\ 1 \\ 1 \\ 1 \end{array}]$ $e^{A t} = e^{3 t} [\begin{array}{r} 1 & 0 & 0 & 0 \\ 6 t & 1 & 0 & 0 \\ 9 t + 18 t^{2} & 6 t & 1 & 0 \\ 12 t + 54 t^{2} + 36 t^{3} & 9 t + 18 t^{2} & 6 t & 1 \end{array}], x (t) = e^{A t} [\begin{array}{r} 1 \\ 1 \\ 1 \\ 1 \end{array}]$
33. $x (t) = [\begin{array}{r} c_{1} \cosh t + c_{2} \sinh t \\ c_{1} \sinh t + c_{2} \cosh t \end{array}]$ $x (t) = [\begin{array}{r} c_{1} \cosh t + c_{2} \sinh t \\ c_{1} \sinh t + c_{2} \cosh t \end{array}]$
35. $e^{A t} = [\begin{array}{r} e^{3 t} & 4 t e^{3 t} \\ 0 & e^{3 t} \end{array}]$ $e^{A t} = [\begin{array}{r} e^{3 t} & 4 t e^{3 t} \\ 0 & e^{3 t} \end{array}]$
36. $e^{A t} = e^{t} [\begin{array}{r} 1 & 2 t & 3 t + 4 t^{2} \\ 0 & 1 & 4 t \\ 0 & 0 & 1 \end{array}]$ $e^{A t} = e^{t} [\begin{array}{r} 1 & 2 t & 3 t + 4 t^{2} \\ 0 & 1 & 4 t \\ 0 & 0 & 1 \end{array}]$
37. $e^{A t} = [\begin{array}{r} e^{2 t} & 3 e^{2 t} - 3 e^{t} & 13 e^{2 t} - (13 + 9 t) e^{t} \\ 0 & e^{t} & 3 t e^{t} \\ 0 & 0 & e^{t} \end{array}]$ $e^{A t} = [\begin{array}{r} e^{2 t} & 3 e^{2 t} - 3 e^{t} & 13 e^{2 t} - (13 + 9 t) e^{t} \\ 0 & e^{t} & 3 t e^{t} \\ 0 & 0 & e^{t} \end{array}]$
38. $e^{A t} = [\begin{array}{r} e^{5 t} & 4 e^{10 t} - 4 e^{5 t} & 16 e^{10 t} - (16 + 50 t) e^{5 t} \\ 0 & e^{10 t} & 4 e^{10 t} - 4 e^{5 t} \\ 0 & 0 & e^{5 t} \end{array}]$ $e^{A t} = [\begin{array}{r} e^{5 t} & 4 e^{10 t} - 4 e^{5 t} & 16 e^{10 t} - (16 + 50 t) e^{5 t} \\ 0 & e^{10 t} & 4 e^{10 t} - 4 e^{5 t} \\ 0 & 0 & e^{5 t} \end{array}]$
39. $e^{A t} = [\begin{array}{r} e^{t} & 3 t e^{t} & 12 e^{2 t} - (12 + 9 t) e^{t} & (51 + 18 t) e^{t} - (51 - 36 t) e^{2 t} \\ 0 & e^{t} & 3 e^{2 t} - 3 e^{t} & 6 e^{t} - (6 - 9 t) e^{2 t} \\ 0 & 0 & e^{2 t} & 4 e^{3 t} - 4 e^{2 t} \\ 0 & 0 & 0 & e^{2 t} \end{array}]$ $e^{A t} = [\begin{array}{r} e^{t} & 3 t e^{t} & 12 e^{2 t} - (12 + 9 t) e^{t} & (51 + 18 t) e^{t} - (51 - 36 t) e^{2 t} \\ 0 & e^{t} & 3 e^{2 t} - 3 e^{t} & 6 e^{t} - (6 - 9 t) e^{2 t} \\ 0 & 0 & e^{2 t} & 4 e^{3 t} - 4 e^{2 t} \\ 0 & 0 & 0 & e^{2 t} \end{array}]$
40. $e^{A t} = [\begin{array}{r} e^{2 t} & 4 t e^{2 t} & (4 t + 8 t^{2}) e^{2 t} & 100 e^{3 t} - (100 + 96 t + 32 t^{2}) e^{2 t} \\ 0 & e^{2 t} & 4 t e^{2 t} & 20 e^{3 t} - (20 + 16 t) e^{2 t} \\ 0 & 0 & e^{2 t} & 4 e^{3 t} - 4 e^{2 t} \\ 0 & 0 & 0 & e^{3 t} \end{array}]$ $e^{A t} = [\begin{array}{r} e^{2 t} & 4 t e^{2 t} & (4 t + 8 t^{2}) e^{2 t} & 100 e^{3 t} - (100 + 96 t + 32 t^{2}) e^{2 t} \\ 0 & e^{2 t} & 4 t e^{2 t} & 20 e^{3 t} - (20 + 16 t) e^{2 t} \\ 0 & 0 & e^{2 t} & 4 e^{3 t} - 4 e^{2 t} \\ 0 & 0 & 0 & e^{3 t} \end{array}]$

..................Content has been hidden....................

You can't read the all page of ebook, please click here login for view all page.

Table of Contents for Section 8.1

Create new playlist

Sign In

Sign Up

Table of Contents for
Section 8.1