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Section 7.6

1. Repeated eigenvalue $λ = - 3$ $λ = - 3$ , eigenvector $v = {[\begin{array}{r} 1 & - 1 \end{array}]}^{T}; x_{1} (t) = (c_{1} + c_{2} + c_{2} t) e^{- 3 t}$ $v = {[\begin{array}{r} 1 & - 1 \end{array}]}^{T}; x_{1} (t) = (c_{1} + c_{2} + c_{2} t) e^{- 3 t}$ , $x_{2} (t) = (- c_{1} - c_{2} t) e^{- 3 t}$ $x_{2} (t) = (- c_{1} - c_{2} t) e^{- 3 t}$
2. Repeated eigenvalue $λ = 2$ $λ = 2$ , single eigenvector $v = {[\begin{array}{r} 1 & 1 \end{array}]}^{T};$ $v = {[\begin{array}{r} 1 & 1 \end{array}]}^{T};$ $x_{1} (t) = (c_{1} + c_{2} + c_{2} t) e^{2 t}, x_{2} (t) = (c_{1} + c_{2} t) e^{2 t}$ $x_{1} (t) = (c_{1} + c_{2} + c_{2} t) e^{2 t}, x_{2} (t) = (c_{1} + c_{2} t) e^{2 t}$
3. Repeated eigenvalue $λ = 3$ $λ = 3$ , eigenvector $v = {[\begin{array}{r} - 2 & 2 \end{array}]}^{T}; x_{1} (t) = (- 2 c_{1} + c_{2} - 2 c_{2} t) e^{3 t}$ $v = {[\begin{array}{r} - 2 & 2 \end{array}]}^{T}; x_{1} (t) = (- 2 c_{1} + c_{2} - 2 c_{2} t) e^{3 t}$ , $x_{2} (t) = (2 c_{1} + 2 c_{2} t) e^{3 t}$ $x_{2} (t) = (2 c_{1} + 2 c_{2} t) e^{3 t}$
4. Repeated eigenvalue $λ = 4$ $λ = 4$ , single eigenvector $v = {[\begin{array}{r} - 1 & 1 \end{array}]}^{T}; x_{1} (t) = (- c_{1} + c_{2} - c_{2} t) e^{4 t}, x_{2} (t) = (c_{1} + c_{2} t) e^{4 t}$ $v = {[\begin{array}{r} - 1 & 1 \end{array}]}^{T}; x_{1} (t) = (- c_{1} + c_{2} - c_{2} t) e^{4 t}, x_{2} (t) = (c_{1} + c_{2} t) e^{4 t}$
5. Repeated eigenvalue $λ = 5$ $λ = 5$ , eigenvector $v = {[\begin{array}{r} 2 & - 4 \end{array}]}^{T}; x_{1} (t) = (2 c_{1} + c_{2} + 2 c_{2} t) e^{5 t}$ $v = {[\begin{array}{r} 2 & - 4 \end{array}]}^{T}; x_{1} (t) = (2 c_{1} + c_{2} + 2 c_{2} t) e^{5 t}$ , $x_{2} (t) = (- 4 c_{1} - 4 c_{2} t) e^{5 t}$ $x_{2} (t) = (- 4 c_{1} - 4 c_{2} t) e^{5 t}$
6. Repeated eigenvalue $λ = 5$ $λ = 5$ , single eigenvector $v = {[\begin{array}{r} - 4 & 4 \end{array}]}^{T}; x_{1} (t) = (- 4 c_{1} + c_{2} - 4 c_{2} t) e^{5 t}, x_{2} (t) = (4 c_{1} + 4 c_{2} t) e^{5 t}$ $v = {[\begin{array}{r} - 4 & 4 \end{array}]}^{T}; x_{1} (t) = (- 4 c_{1} + c_{2} - 4 c_{2} t) e^{5 t}, x_{2} (t) = (4 c_{1} + 4 c_{2} t) e^{5 t}$
7. Eigenvalues $λ = 2$ $λ = 2$ , 2, 9 with three linearly independent eigenvectors; $x_{1} (t) = c_{1} e^{2 t} + c_{2} e^{2 t}$ $x_{1} (t) = c_{1} e^{2 t} + c_{2} e^{2 t}$ , $x_{2} (t) = c_{1} e^{2 t} + c_{3} e^{9 t}, x_{3} (t) = c_{2} e^{2 t}$ $x_{2} (t) = c_{1} e^{2 t} + c_{3} e^{9 t}, x_{3} (t) = c_{2} e^{2 t}$
8. Eigenvalues $λ = 7$ $λ = 7$ , 13, 13 with three linearly independent eigenvectors; $x_{1} (t) = 2 c_{1} e^{7 t} - c_{3} e^{13 t}, x_{2} (t) = - 3 c_{1} e^{7 t} + c_{3} e^{13 t}, x_{3} (t) = c_{1} e^{7 t} + c_{2} e^{13 t}$ $x_{1} (t) = 2 c_{1} e^{7 t} - c_{3} e^{13 t}, x_{2} (t) = - 3 c_{1} e^{7 t} + c_{3} e^{13 t}, x_{3} (t) = c_{1} e^{7 t} + c_{2} e^{13 t}$
9. Eigenvalues $λ = 5$ $λ = 5$ , 5, 9 with three linearly independent eigenvectors; $x_{1} (t) = c_{1} e^{5 t} + 7 c_{2} e^{5 t} + 3 c_{3} e^{9 t}, x_{2} (t) = 2 c_{1} e^{5 t}, x_{3} (t) = 2 c_{2} e^{5 t} + c_{3} e^{9 t}$ $x_{1} (t) = c_{1} e^{5 t} + 7 c_{2} e^{5 t} + 3 c_{3} e^{9 t}, x_{2} (t) = 2 c_{1} e^{5 t}, x_{3} (t) = 2 c_{2} e^{5 t} + c_{3} e^{9 t}$
10. Eigenvalues $λ = 3$ $λ = 3$ , 3, 7 with three linearly independent eigenvectors; $x_{1} (t) = 5 c_{1} e^{3 t} - 3 c_{2} e^{3 t} + 2 c_{3} e^{7 t}$ $x_{1} (t) = 5 c_{1} e^{3 t} - 3 c_{2} e^{3 t} + 2 c_{3} e^{7 t}$ , $x_{2} (t) = 2 c_{1} e^{3 t} + c_{3} e^{7 t}, x_{3} (t) = c_{2} e^{3 t}$ $x_{2} (t) = 2 c_{1} e^{3 t} + c_{3} e^{7 t}, x_{3} (t) = c_{2} e^{3 t}$
11. Triple eigenvalue $λ = - 1$ $λ = - 1$ of defect 2; $x_{1} (t) = (- 2 c_{2} + c_{3} - 2 c_{3} t) e^{- t}, x_{2} (t) = (c_{1} - c_{2} + c_{2} t - c_{3} t + \frac{1}{2} c_{3} t^{2}) e^{- t}, x_{3} (t) = (c_{2} + c_{3} t) e^{- t}$ $x_{1} (t) = (- 2 c_{2} + c_{3} - 2 c_{3} t) e^{- t}, x_{2} (t) = (c_{1} - c_{2} + c_{2} t - c_{3} t + \frac{1}{2} c_{3} t^{2}) e^{- t}, x_{3} (t) = (c_{2} + c_{3} t) e^{- t}$
12. Triple eigenvalue $λ = - 1$ $λ = - 1$ of defect 2; $x_{1} (t) = e^{- t} (c_{1} + c_{3} + c_{2} t + \frac{1}{2} c_{3} t^{2}) x_{2} (t) = e^{- t} (c_{1} + c_{2} t + \frac{1}{2} c_{3} t^{2}), x_{3} (t) = e^{- t} (c_{2} + c_{3} t)$ $x_{1} (t) = e^{- t} (c_{1} + c_{3} + c_{2} t + \frac{1}{2} c_{3} t^{2}) x_{2} (t) = e^{- t} (c_{1} + c_{2} t + \frac{1}{2} c_{3} t^{2}), x_{3} (t) = e^{- t} (c_{2} + c_{3} t)$
13. Triple eigenvalue $λ = - 1$ $λ = - 1$ of defect 2; $x_{1} (t) = (c_{1} + c_{2} t + \frac{1}{2} c_{3} t^{2}) e^{- t}, x_{2} (t) = (2 c_{2} + c_{3} + 2 c_{3} t) e^{- t}$ $x_{1} (t) = (c_{1} + c_{2} t + \frac{1}{2} c_{3} t^{2}) e^{- t}, x_{2} (t) = (2 c_{2} + c_{3} + 2 c_{3} t) e^{- t}$ , $x_{3} (t) = (c_{2} + c_{3} t) e^{- t}$ $x_{3} (t) = (c_{2} + c_{3} t) e^{- t}$
14. Triple eigenvalue $λ = - 1$ $λ = - 1$ of defect 2; $x_{1} (t) = e^{- t} (5 c_{1} + c_{2} + c_{3} + 5 c_{2} t + c_{3} t + \frac{5}{2} c_{3} t^{2}), x_{2} (t) = e^{- t} (- 25 c_{1} - 5 c_{2} - 25 c_{2} t - 5 c_{3} t - \frac{25}{2} c_{3} t^{2}), x_{3} (t) = e^{- t} (- 5 c_{1} + 4 c_{2} - 5 c_{2} t + 4 c_{3} t - \frac{5}{2} c_{3} t^{2})$ $x_{1} (t) = e^{- t} (5 c_{1} + c_{2} + c_{3} + 5 c_{2} t + c_{3} t + \frac{5}{2} c_{3} t^{2}), x_{2} (t) = e^{- t} (- 25 c_{1} - 5 c_{2} - 25 c_{2} t - 5 c_{3} t - \frac{25}{2} c_{3} t^{2}), x_{3} (t) = e^{- t} (- 5 c_{1} + 4 c_{2} - 5 c_{2} t + 4 c_{3} t - \frac{5}{2} c_{3} t^{2})$
15. Triple eigenvalue $λ = 1$ $λ = 1$ of defect 1; $x_{1} (t) = (3 c_{1} + c_{3} - 3 c_{3} t) e^{t}, x_{2} (t) = (- c_{1} + c_{3} t) e^{t}, x_{3} (t) = (c_{2} + c_{3} t) e^{t}$ $x_{1} (t) = (3 c_{1} + c_{3} - 3 c_{3} t) e^{t}, x_{2} (t) = (- c_{1} + c_{3} t) e^{t}, x_{3} (t) = (c_{2} + c_{3} t) e^{t}$
16. Triple eigenvalue $λ = 1$ $λ = 1$ of defect 1; $x_{1} (t) = e^{t} (3 c_{1} + 3 c_{2} + c_{3})$ $x_{1} (t) = e^{t} (3 c_{1} + 3 c_{2} + c_{3})$ $x_{2} (t) = e^{t} (- 2 c_{1} - 2 c_{3} t)$ $x_{2} (t) = e^{t} (- 2 c_{1} - 2 c_{3} t)$ , $x_{3} (t) = e^{t} (- 2 c_{2} + 2 c_{3} t)$ $x_{3} (t) = e^{t} (- 2 c_{2} + 2 c_{3} t)$
17. Triple eigenvalue $λ = 1$ $λ = 1$ of defect 1; $x_{1} (t) = (2 c_{1} + c_{2}) e^{t}$ $x_{1} (t) = (2 c_{1} + c_{2}) e^{t}$ , $x_{2} (t) = (- 3 c_{2} + c_{3} + 6 c_{3} t) e^{t}$ $x_{2} (t) = (- 3 c_{2} + c_{3} + 6 c_{3} t) e^{t}$ , $x_{3} (t) = - 9 (c_{1} + c_{3} t) e^{t}$ $x_{3} (t) = - 9 (c_{1} + c_{3} t) e^{t}$
18. Triple eigenvalue $λ = 1$ $λ = 1$ of defect 1; $x_{1} (t) = e^{t} (- c_{1} - 2 c_{2} + c_{3})$ $x_{1} (t) = e^{t} (- c_{1} - 2 c_{2} + c_{3})$ , $x_{2} (t) = e^{t} (c_{2} + c_{3} t)$ $x_{2} (t) = e^{t} (c_{2} + c_{3} t)$ , $x_{3} (t) = e^{t} (c_{1} - 2 c_{3} t)$ $x_{3} (t) = e^{t} (c_{1} - 2 c_{3} t)$
19. Double eigenvalues $λ = - 1$ $λ = - 1$ and $λ = 1$ $λ = 1$ , with four linearly independent solutions; $x_{1} (t) = c_{1} e^{- t} + c_{4} e^{t}$ $x_{1} (t) = c_{1} e^{- t} + c_{4} e^{t}$ , $x_{2} (t) = c_{3} e^{t}, x_{3} (t) = c_{2} e^{- t} + 3 c_{4} e^{t}$ $x_{2} (t) = c_{3} e^{t}, x_{3} (t) = c_{2} e^{- t} + 3 c_{4} e^{t}$ , $x_{4} (t) = c_{1} e^{- t} - 2 c_{3} e^{t}$ $x_{4} (t) = c_{1} e^{- t} - 2 c_{3} e^{t}$
20. Eigenvalue $λ = 2$ $λ = 2$ with multiplicity 4 and defect 3; $x_{1} (t) = (c_{1} + c_{3} + c_{2} t + c_{4} t + \frac{1}{2} c_{3} t^{2} + \frac{1}{6} c_{4} t^{3}) e^{2 t}$ $x_{1} (t) = (c_{1} + c_{3} + c_{2} t + c_{4} t + \frac{1}{2} c_{3} t^{2} + \frac{1}{6} c_{4} t^{3}) e^{2 t}$ , $x_{2} (t) = (c_{2} + c_{3} t + \frac{1}{2} c_{4} t^{2}) e^{2 t}$ $x_{2} (t) = (c_{2} + c_{3} t + \frac{1}{2} c_{4} t^{2}) e^{2 t}$ , $x_{3} (t) = (c_{3} + c_{4} t) e^{2 t}$ $x_{3} (t) = (c_{3} + c_{4} t) e^{2 t}$ , $x_{4} (t) = c_{4} e^{2 t}$ $x_{4} (t) = c_{4} e^{2 t}$
21. Eigenvalue $λ = 1$ $λ = 1$ with multiplicity 4 and defect 2; $x_{1} (t) = (- 2 c_{2} + c_{3} - 2 c_{3} t) e^{t}, x_{2} (t) = (c_{2} + c_{3} t) e^{t}$ $x_{1} (t) = (- 2 c_{2} + c_{3} - 2 c_{3} t) e^{t}, x_{2} (t) = (c_{2} + c_{3} t) e^{t}$ , $x_{3} (t) = (c_{2} + c_{4} + c_{3} t) e^{t}, x_{4} (t) = (c_{1} + c_{2} t + \frac{1}{2} c_{3} t^{2}) e^{t}$ $x_{3} (t) = (c_{2} + c_{4} + c_{3} t) e^{t}, x_{4} (t) = (c_{1} + c_{2} t + \frac{1}{2} c_{3} t^{2}) e^{t}$
22. Eigenvalue $λ = 1$ $λ = 1$ with multiplicity 4 and defect 2; $x_{1} (t) = (c_{1} + 3 c_{2} + c_{4} + c_{2} t + 3 c_{3} t + \frac{1}{2} c_{3} t^{2}) e^{t}$ $x_{1} (t) = (c_{1} + 3 c_{2} + c_{4} + c_{2} t + 3 c_{3} t + \frac{1}{2} c_{3} t^{2}) e^{t}$ , $x_{2} (t) = - (2 c_{2} - c_{3} + 2 c_{3} t) e^{t}, x_{3} (t) = (c_{2} + c_{3} t) e^{t}$ $x_{2} (t) = - (2 c_{2} - c_{3} + 2 c_{3} t) e^{t}, x_{3} (t) = (c_{2} + c_{3} t) e^{t}$ , $x_{4} (t) = - (2 c_{1} + 6 c_{2} + 2 c_{2} t + 6 c_{3} t + c_{3} t^{2}) e^{t}$ $x_{4} (t) = - (2 c_{1} + 6 c_{2} + 2 c_{2} t + 6 c_{3} t + c_{3} t^{2}) e^{t}$
23. $x (t) = c_{1} v_{1} e^{- t} + (c_{2} v_{2} + c_{3} v_{3}) e^{3 t}$ $x (t) = c_{1} v_{1} e^{- t} + (c_{2} v_{2} + c_{3} v_{3}) e^{3 t}$ with $v_{1} = {[\begin{array}{r} 1 & - 1 & 2 \end{array}]}^{T}, v_{2} = {[\begin{array}{r} 4 & 0 & 9 \end{array}]}^{T}, v_{3} = {[\begin{array}{r} 0 & 2 & 1 \end{array}]}^{T}$ $v_{1} = {[\begin{array}{r} 1 & - 1 & 2 \end{array}]}^{T}, v_{2} = {[\begin{array}{r} 4 & 0 & 9 \end{array}]}^{T}, v_{3} = {[\begin{array}{r} 0 & 2 & 1 \end{array}]}^{T}$
24. $x (t) = c_{1} v_{1} e^{- t} + (c_{2} v_{2} + c_{3} v_{3}) e^{3 t}$ $x (t) = c_{1} v_{1} e^{- t} + (c_{2} v_{2} + c_{3} v_{3}) e^{3 t}$ with $v_{1} = {[\begin{array}{r} 5 & 3 & - 3 \end{array}]}^{T}, v_{2} = {[\begin{array}{r} 4 & 0 & - 1 \end{array}]}^{T}, v_{3} = {[\begin{array}{r} 2 & - 1 & 0 \end{array}]}^{T}$ $v_{1} = {[\begin{array}{r} 5 & 3 & - 3 \end{array}]}^{T}, v_{2} = {[\begin{array}{r} 4 & 0 & - 1 \end{array}]}^{T}, v_{3} = {[\begin{array}{r} 2 & - 1 & 0 \end{array}]}^{T}$
25. $x (t) = [c_{1} v_{1} + c_{2} (v_{1} t + v_{2}) + c_{3} (\frac{1}{2} v_{1} t^{2} + v_{2} t + v_{3})] e^{2 t}$ $x (t) = [c_{1} v_{1} + c_{2} (v_{1} t + v_{2}) + c_{3} (\frac{1}{2} v_{1} t^{2} + v_{2} t + v_{3})] e^{2 t}$ with $v_{1} = {[\begin{array}{r} - 1 & 0 & - 1 \end{array}]}^{T}, v_{2} = {[\begin{array}{r} - 4 & - 1 & 0 \end{array}]}^{T},$ $v_{1} = {[\begin{array}{r} - 1 & 0 & - 1 \end{array}]}^{T}, v_{2} = {[\begin{array}{r} - 4 & - 1 & 0 \end{array}]}^{T},$ and $v_{3} = {[\begin{array}{r} 1 & 0 & 0 \end{array}]}^{T}$ $v_{3} = {[\begin{array}{r} 1 & 0 & 0 \end{array}]}^{T}$
26. $x (t) = [c_{1} v_{1} + c_{2} (v_{1} t + v_{2}) + c_{3} (\frac{1}{2} v_{1} t^{2} + v_{2} t + v_{3})] e^{3 t}$ $x (t) = [c_{1} v_{1} + c_{2} (v_{1} t + v_{2}) + c_{3} (\frac{1}{2} v_{1} t^{2} + v_{2} t + v_{3})] e^{3 t}$ with $v_{1} = {[\begin{array}{r} 0 & 2 & 2 \end{array}]}^{T}, v_{3} = {[\begin{array}{r} 2 & 1 & - 3 \end{array}]}^{T},$ $v_{1} = {[\begin{array}{r} 0 & 2 & 2 \end{array}]}^{T}, v_{3} = {[\begin{array}{r} 2 & 1 & - 3 \end{array}]}^{T},$ and $v_{3} = {[\begin{array}{r} 1 & 0 & 0 \end{array}]}^{T}$ $v_{3} = {[\begin{array}{r} 1 & 0 & 0 \end{array}]}^{T}$
27. $x (t) = [c_{1} v_{1} + c_{2} (v_{1} t + v_{2}) + c_{3} v_{3}] e^{2 t}$ $x (t) = [c_{1} v_{1} + c_{2} (v_{1} t + v_{2}) + c_{3} v_{3}] e^{2 t}$ with $v_{1} = {[\begin{array}{r} - 5 & 3 & 8 \end{array}]}^{T}, v_{2} = {[\begin{array}{r} 1 & 0 & 0 \end{array}]}^{T}, v_{3} = {[\begin{array}{r} 1 & 0 & 0 \end{array}]}^{T}$ $v_{1} = {[\begin{array}{r} - 5 & 3 & 8 \end{array}]}^{T}, v_{2} = {[\begin{array}{r} 1 & 0 & 0 \end{array}]}^{T}, v_{3} = {[\begin{array}{r} 1 & 0 & 0 \end{array}]}^{T}$
28. $x (t) = [c_{1} v_{1} + c_{2} (v_{1} t + v_{2}) + c_{3} (\frac{1}{2} v_{1} t^{2} + v_{2} t + v_{3})] e^{2 t}$ $x (t) = [c_{1} v_{1} + c_{2} (v_{1} t + v_{2}) + c_{3} (\frac{1}{2} v_{1} t^{2} + v_{2} t + v_{3})] e^{2 t}$ with $v_{1} = {[\begin{array}{r} 119 & - 289 & 0 \end{array}]}^{T}, v_{2} = {[\begin{array}{r} - 17 & 34 & 17 \end{array}]}^{T},$ $v_{1} = {[\begin{array}{r} 119 & - 289 & 0 \end{array}]}^{T}, v_{2} = {[\begin{array}{r} - 17 & 34 & 17 \end{array}]}^{T},$ and $v_{3} = {[\begin{array}{r} 1 & 0 & 0 \end{array}]}^{T}$ $v_{3} = {[\begin{array}{r} 1 & 0 & 0 \end{array}]}^{T}$
29. $x (t) = [c_{1} v_{1} + c_{2} (v_{1} t + v_{2})] e^{- t} + [c_{3} v_{3} + c_{4} (v_{3} t + v_{4})] e^{2 t}$ $x (t) = [c_{1} v_{1} + c_{2} (v_{1} t + v_{2})] e^{- t} + [c_{3} v_{3} + c_{4} (v_{3} t + v_{4})] e^{2 t}$ with $v_{1} = {[\begin{array}{r} 1 & - 3 & - 1 & - 2 \end{array}]}^{T}, v_{2} = {[\begin{array}{r} 0 & 1 & 0 & 0 \end{array}]}^{T}, v_{3} = {[\begin{array}{r} 0 & - 1 & 1 & 0 \end{array}]}^{T}, v_{4} = {[\begin{array}{r} 0 & 0 & 2 & 1 \end{array}]}^{T}$ $v_{1} = {[\begin{array}{r} 1 & - 3 & - 1 & - 2 \end{array}]}^{T}, v_{2} = {[\begin{array}{r} 0 & 1 & 0 & 0 \end{array}]}^{T}, v_{3} = {[\begin{array}{r} 0 & - 1 & 1 & 0 \end{array}]}^{T}, v_{4} = {[\begin{array}{r} 0 & 0 & 2 & 1 \end{array}]}^{T}$
30. $x (t) = [c_{1} v_{1} + c_{2} (v_{1} t + v_{2})] e^{- t} + [c_{3} v_{3} + c_{4} (v_{3} t + v_{4})] e^{2 t}$ $x (t) = [c_{1} v_{1} + c_{2} (v_{1} t + v_{2})] e^{- t} + [c_{3} v_{3} + c_{4} (v_{3} t + v_{4})] e^{2 t}$ , with $v_{1} = {[\begin{array}{r} 0 & 1 & - 1 & - 3 \end{array}]}^{T}, v_{2} = {[\begin{array}{r} 0 & 0 & 1 & 2 \end{array}]}^{T}, v_{3} = {[\begin{array}{r} - 1 & 0 & 0 & 0 \end{array}]}^{T}, v_{4} = {[\begin{array}{r} 0 & 0 & 3 & 5 \end{array}]}^{T}$ $v_{1} = {[\begin{array}{r} 0 & 1 & - 1 & - 3 \end{array}]}^{T}, v_{2} = {[\begin{array}{r} 0 & 0 & 1 & 2 \end{array}]}^{T}, v_{3} = {[\begin{array}{r} - 1 & 0 & 0 & 0 \end{array}]}^{T}, v_{4} = {[\begin{array}{r} 0 & 0 & 3 & 5 \end{array}]}^{T}$
31. $x (t) = [c_{1} v_{1} + c_{2} (v_{1} t + v_{2}) + c_{3} (\frac{1}{2} v_{1} t^{2} + v_{2} t + v_{3}) + c_{4} v_{4}] e^{t}$ $x (t) = [c_{1} v_{1} + c_{2} (v_{1} t + v_{2}) + c_{3} (\frac{1}{2} v_{1} t^{2} + v_{2} t + v_{3}) + c_{4} v_{4}] e^{t}$ with $v_{1} = {[\begin{array}{r} 42 & 7 & - 21 & - 42 \end{array}]}^{T}, v_{2} = {[\begin{array}{r} 34 & 22 & - 10 & - 27 \end{array}]}^{T}, v_{3} = {[\begin{array}{r} 1 & 0 & 0 & 0 \end{array}]}^{T}, v_{4} = {[\begin{array}{r} 0 & 1 & 3 & 0 \end{array}]}^{T}$ $v_{1} = {[\begin{array}{r} 42 & 7 & - 21 & - 42 \end{array}]}^{T}, v_{2} = {[\begin{array}{r} 34 & 22 & - 10 & - 27 \end{array}]}^{T}, v_{3} = {[\begin{array}{r} 1 & 0 & 0 & 0 \end{array}]}^{T}, v_{4} = {[\begin{array}{r} 0 & 1 & 3 & 0 \end{array}]}^{T}$
32. $x (t) = (c_{1} v_{1} + c_{2} v_{2}) e^{2 t} + (c_{3} v_{3} + c_{4} v_{4} + c_{5} v_{5}) e^{3 t}$ $x (t) = (c_{1} v_{1} + c_{2} v_{2}) e^{2 t} + (c_{3} v_{3} + c_{4} v_{4} + c_{5} v_{5}) e^{3 t}$ with $v_{1} = {[\begin{array}{r} 8 & 0 & - 3 & 1 & 0 \end{array}]}^{T}, v_{2} = {[\begin{array}{r} 1 & 0 & 0 & 0 & 3 \end{array}]}^{T}, v_{3} = {[\begin{array}{r} 3 & - 2 & - 1 & 0 & 0 \end{array}]}^{T}, v_{4} = {[\begin{array}{r} 2 & - 2 & 0 & - 3 & 0 \end{array}]}^{T}, v_{5} = {[\begin{array}{r} 1 & - 1 & 0 & 0 & 3 \end{array}]}^{T}$ $v_{1} = {[\begin{array}{r} 8 & 0 & - 3 & 1 & 0 \end{array}]}^{T}, v_{2} = {[\begin{array}{r} 1 & 0 & 0 & 0 & 3 \end{array}]}^{T}, v_{3} = {[\begin{array}{r} 3 & - 2 & - 1 & 0 & 0 \end{array}]}^{T}, v_{4} = {[\begin{array}{r} 2 & - 2 & 0 & - 3 & 0 \end{array}]}^{T}, v_{5} = {[\begin{array}{r} 1 & - 1 & 0 & 0 & 3 \end{array}]}^{T}$
33. $x_{1} (t) = {[\begin{array}{r} \cos 4 t & \sin 4 t & 0 & 0 \end{array}]}^{T} e^{3 t}, x_{2} (t) = {[\begin{array}{r} - \sin 4 t & \cos 4 t & 0 & 0 \end{array}]}^{T} e^{3 t}, x_{3} (t) = {[\begin{array}{r} t \cos 4 t & t \sin 4 t & \cos 4 t & \sin 4 t \end{array}]}^{T} e^{3 t}, x_{4} (t) = {[- \begin{array}{r} t \sin 4 t & t \cos 4 t & - \sin 4 t & \cos 4 t \end{array}]}^{T} e^{3 t}$ $x_{1} (t) = {[\begin{array}{r} \cos 4 t & \sin 4 t & 0 & 0 \end{array}]}^{T} e^{3 t}, x_{2} (t) = {[\begin{array}{r} - \sin 4 t & \cos 4 t & 0 & 0 \end{array}]}^{T} e^{3 t}, x_{3} (t) = {[\begin{array}{r} t \cos 4 t & t \sin 4 t & \cos 4 t & \sin 4 t \end{array}]}^{T} e^{3 t}, x_{4} (t) = {[- \begin{array}{r} t \sin 4 t & t \cos 4 t & - \sin 4 t & \cos 4 t \end{array}]}^{T} e^{3 t}$
34. $x_{1} (t) = [\begin{array}{c} \sin 3 t \\ 3 \cos 3 t - 3 \cos 3 t \\ \begin{array}{c} 0 \\ \sin 3 t \end{array} \end{array}] e^{2 t}, x_{2} (t) = [\begin{array}{c} - \cos 3 t \\ 3 \cos 3 t + 3 \cos 3 t \\ \begin{array}{c} 0 \\ - \cos 3 t \end{array} \end{array}] e^{2 t}, x_{3} (t) = [\begin{array}{c} 3 \cos 3 t + t \sin 3 t \\ (3 t - 10) \cos 3 t - (3 t + 9) \sin 3 t \\ \sin 3 t \\ t \sin 3 t \end{array}] e^{2 t}, x_{4} (t) = [\begin{array}{c} - t \cos 3 t + t \sin 3 t \\ (3 t + 9) \cos 3 t + (3 t - 10) \sin 3 t \\ - \cos 3 t \\ - t \cos 3 t \end{array}] e^{2 t}$ $x_{1} (t) = [\begin{array}{c} \sin 3 t \\ 3 \cos 3 t - 3 \cos 3 t \\ \begin{array}{c} 0 \\ \sin 3 t \end{array} \end{array}] e^{2 t}, x_{2} (t) = [\begin{array}{c} - \cos 3 t \\ 3 \cos 3 t + 3 \cos 3 t \\ \begin{array}{c} 0 \\ - \cos 3 t \end{array} \end{array}] e^{2 t}, x_{3} (t) = [\begin{array}{c} 3 \cos 3 t + t \sin 3 t \\ (3 t - 10) \cos 3 t - (3 t + 9) \sin 3 t \\ \sin 3 t \\ t \sin 3 t \end{array}] e^{2 t}, x_{4} (t) = [\begin{array}{c} - t \cos 3 t + t \sin 3 t \\ (3 t + 9) \cos 3 t + (3 t - 10) \sin 3 t \\ - \cos 3 t \\ - t \cos 3 t \end{array}] e^{2 t}$
35. $x_{1} (t) = x_{2} (t) = v_{0} (1 - e^{- t})$ $x_{1} (t) = x_{2} (t) = v_{0} (1 - e^{- t})$ ; $\lim_{t \to \infty} x_{1} (t) = \lim_{t \to \infty} x_{2} (t) = v_{0}$ $\lim_{t \to \infty} x_{1} (t) = \lim_{t \to \infty} x_{2} (t) = v_{0}$
36. $x_{1} (t) = v_{0} (2 - 2 e^{- t} - t e^{- t}), x_{2} (t) = v_{0} (2 - 2 e^{- t} - t e^{- t} - \frac{1}{2} t^{2} e^{- t})$ $x_{1} (t) = v_{0} (2 - 2 e^{- t} - t e^{- t}), x_{2} (t) = v_{0} (2 - 2 e^{- t} - t e^{- t} - \frac{1}{2} t^{2} e^{- t})$ ; $\lim_{t \to \infty} x_{1} (t) = \lim_{t \to \infty} x_{2} (t) = 2 v_{0}$ $\lim_{t \to \infty} x_{1} (t) = \lim_{t \to \infty} x_{2} (t) = 2 v_{0}$

In Problems 37, 39, 41, 43, and 45 we give a nonsingular matrix Q and a Jordan-form matrix J such that $A = Q J Q^{- 1}$ $A = Q J Q^{- 1}$ . Any scalar multiple of Q will do the same job.

37. $Q = [\begin{array}{r} 1 & 4 & - 2 \\ - 1 & 0 & 9 \\ 2 & 9 & 0 \end{array}], J = [\begin{array}{r} - 1 & 0 & - 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{array}]$ $Q = [\begin{array}{r} 1 & 4 & - 2 \\ - 1 & 0 & 9 \\ 2 & 9 & 0 \end{array}], J = [\begin{array}{r} - 1 & 0 & - 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{array}]$
39. $Q = [\begin{array}{r} 1 & 4 & - 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array}], J = [\begin{array}{r} 2 & 1 & 0 \\ 0 & 2 & 1 \\ 0 & 0 & 2 \end{array}]$ $Q = [\begin{array}{r} 1 & 4 & - 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array}], J = [\begin{array}{r} 2 & 1 & 0 \\ 0 & 2 & 1 \\ 0 & 0 & 2 \end{array}]$
41. $Q = [\begin{array}{r} 0 & - 5 & 0 \\ 1 & 3 & 0 \\ 1 & 8 & 1 \end{array}], J = [\begin{array}{r} 2 & 0 & 0 \\ 0 & 2 & 1 \\ 0 & 0 & 2 \end{array}]$ $Q = [\begin{array}{r} 0 & - 5 & 0 \\ 1 & 3 & 0 \\ 1 & 8 & 1 \end{array}], J = [\begin{array}{r} 2 & 0 & 0 \\ 0 & 2 & 1 \\ 0 & 0 & 2 \end{array}]$
43. $Q = [\begin{array}{r} - 1 & 0 & 0 & 0 \\ 3 & - 1 & - 1 & 2 \\ 1 & 0 & 1 & 0 \\ 2 & 0 & 0 & 1 \end{array}], J = [\begin{array}{r} - 1 & 1 & 0 & 0 \\ 0 & - 1 & 0 & 0 \\ 0 & 0 & 2 & 1 \\ 0 & 0 & 0 & 2 \end{array}]$ $Q = [\begin{array}{r} - 1 & 0 & 0 & 0 \\ 3 & - 1 & - 1 & 2 \\ 1 & 0 & 1 & 0 \\ 2 & 0 & 0 & 1 \end{array}], J = [\begin{array}{r} - 1 & 1 & 0 & 0 \\ 0 & - 1 & 0 & 0 \\ 0 & 0 & 2 & 1 \\ 0 & 0 & 0 & 2 \end{array}]$
45. $Q = [\begin{array}{r} - 30 & 432 & 360 & 0 \\ - 7 & 72 & 228 & 0 \\ 9 & - 216 & - 108 & 0 \\ 30 & - 432 & - 288 & 12 \end{array}], J = [\begin{array}{r} 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{array}]$ $Q = [\begin{array}{r} - 30 & 432 & 360 & 0 \\ - 7 & 72 & 228 & 0 \\ 9 & - 216 & - 108 & 0 \\ 30 & - 432 & - 288 & 12 \end{array}], J = [\begin{array}{r} 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{array}]$

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Table of Contents for Section 7.6

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Table of Contents for
Section 7.6