1. 

1. (a) $\left[\begin{array}{cc}\hfill c_1{e}^{2t}& +c_2{e}^{3t}\hfill \\ \hfill c_1{e}^{2t}& +2c_2{e}^{3t}\hfill \end{array}\right]$;

2. (b) $\left[\begin{array}{cc}\hfill -c_1{e}^{-2t}& -4c_2{e}^t\hfill \\ \hfill c_1{e}^{-2t}& +c_2{e}^t\hfill \end{array}\right]$;

3. (c) $\left[\begin{array}{cc}\hfill 2c_1& +c_2{e}^{5t}\hfill \\ \hfill c_1& -2c_2{e}^{5t}\hfill \end{array}\right]$;

4. (d) $\left[\begin{array}{r}-c_1{e}^t\sin t+c_2{e}^t\text{ cos }t\\ c_1{e}^t\text{ cos }t+c_2{e}^t\sin t\end{array}\right]$;

5. (e) $\left[\begin{array}{r}-c_1{e}^{3t}\mathrm{sin }2t+c_2{e}^{3t}\text{ cos }2t\\ c_1{e}^{3t}\text{ cos }2t+c_2{e}^{3t}\mathrm{sin }2t\end{array}\right]$;

6. (f) $\left[\begin{array}{cc}\hfill -c_1& +c_2{e}^{5t}+c_3{e}^t\hfill \\ \hfill -3c_1& +8c_2{e}^{5t}\hfill \\ \hfill c_1& +4c_2{e}^{5t}\hfill \end{array}\right]$

2. 

1. (a) $\left[\begin{array}{r}{e}^{-3t}+2e^t\\ -e^{-3t}+2e^t\end{array}\right]$;

2. (b) $\left[\begin{array}{r}{e}^t\text{ cos }2t+2e^t\text{ sin }2t\\ e^t\text{ sin }2t-2e^t\text{ cos }2t\end{array}\right]$;

3. (c) $\left[\begin{array}{cc}\hfill -6e^t& +2e^{-t}+6\hfill \\ \hfill -3e^t& +e^{-t}+4\hfill \\ \hfill -e^t& +e^{-t}+2\hfill \end{array}\right]$;

4. (d) $\left[\begin{array}{c}-2-3e^t+6e^{2t}\hfill \\ \phantom{-}1+3e^t-3e^t\hfill \\ \phantom{-}1+3e^t\hfill \end{array}\right]$

4. $\begin{array}{c}{y}_1(t)=15e^{-0.24t}+25e^{-0.08t}\end{array},\begin{array}{c}{y}_2(t)=-30e^{-0.24t}+50e^{-0.08t}\end{array}$

5. 

1. (a) $\left[\begin{array}{r}-2c_1{e}^t-2c_2{e}^{-t}+c_3{e}^{\sqrt{2}t}+c_4{e}^{-\sqrt{2}t}\\ c_1{e}^t+c_2{e}^{-t}-c_3{e}^{\sqrt{2}t}-c_4{e}^{-\sqrt{2}t}\end{array}\right]$

2. (b) $\left[\begin{array}{r}{c}_1{e}^{2t}+c_2{e}^{-2t}-c_3{e}^t-c_4{e}^{-t}\\ c_1{e}^{2t}-c_2{e}^{-2t}+c_3{e}^t-c_4{e}^{-t}\end{array}\right]$

6. $\begin{array}{c}\begin{array}{c}{y}_1(t)=-e^{2t}+e^{-2t}+e^t;\end{array}\\ y_2(t)=-e^{2t}-e^{-2t}+2e^t\\ \end{array}$

8. $\begin{array}{c}{x}_1(t)=\text{cos }t+3\sin t+\frac{1}{\sqrt{3}}\sin \sqrt{3}t,\\ x_2(t)=\text{cos }t+3\sin t-\frac{1}{\sqrt{3}}\sin \sqrt{3}t\\ \end{array}$

10. 

1. (a)
   $\begin{array}{ccc}\hfill m_1{x}_1^{″}(t)& =& -{\mathit{\text{kx}}}_1+k(x_2-x_1)\hfill \\ \hfill m_2{x}_2^{″}(t)& =& -k(x_2-x_1)+k(x_3-x_2)\hfill \\ \hfill m_3{x}_3^{″}(t)& =& -k(x_3-x_2)-{\mathit{\text{kx}}}_3\hfill \end{array}$

2. (b) $\left[\begin{array}{r}\begin{array}{c}\hfill 0.1\text{ cos }2\sqrt{3}t+0.9\text{ cos }\sqrt{2}t\\ -0.2\text{ cos }2\sqrt{3}t+1.2\text{ cos }\sqrt{2}t\end{array}\\ 0.1\text{ cos }2\sqrt{3}t+0.9\text{ cos }\sqrt{2}t\end{array}\right]$

11. $p(\lambda )={(-1)}^n({\lambda }^n-a_{n-1}{\lambda }^{n-1}-\cdots -a_1\lambda -a_0)$