Linear Algebra with Applications, 10th Edition

Section 6.2 Exercises

1. Find the general solution of each of the following systems:

   1. $\begin{array}{ccccc}\hfill y_1^{\prime }& \hfill =\hfill & \hfill y_1& \hfill +\hfill & \hfill y_2\\ \hfill y_2^{\prime }& \hfill =\hfill & \hfill -2y_1& \hfill +\hfill & \hfill 4y_2\end{array}$

   2. $\begin{array}{ccccc}\hfill y_1^{\prime }& \hfill =\hfill & \hfill {2y}_1& \hfill +\hfill & \hfill {4y}_2\\ \hfill y_2^{\prime }& \hfill =\hfill & \hfill -y_1& -& \hfill 3y_2\end{array}$

   3. $\begin{array}{ccccc}\hfill y_1^{\prime }& \hfill =\hfill & \hfill y_1& -& \hfill {2y}_2\\ \hfill y_2^{\prime }& \hfill =\hfill & \hfill -{2y}_1& +& \hfill 4y_2\end{array}$

   4. $\begin{array}{ccccc}\hfill y_1^{\prime }& \hfill =\hfill & \hfill y_1& -& \hfill y_2\\ \hfill y_2^{\prime }& \hfill =\hfill & y_1& +& \hfill y_2\end{array}$

   5. $\begin{array}{ccccc}\hfill y_1^{\prime }& \hfill =\hfill & \hfill {3y}_1& -& \hfill {2y}_2\\ \hfill y_2^{\prime }& \hfill =\hfill & {2y}_1& +& \hfill {3y}_2\end{array}$

   6. $\begin{array}{ccccc}\hfill y_1^{\prime }& =& \hfill y_1& \hfill +\hfill & \hfill y_3\\ \hfill y_2^{\prime }& =& \hfill 2y_2& \hfill +\hfill & \hfill 6y_3\\ \hfill y_3^{\prime }& =& \hfill y_2& \hfill +\hfill & \hfill 3y_3\end{array}$

2. Solve each of the following initial value problems:

   1. $\begin{array}{c}{y}_1^{\prime }=\phantom{2}-y_1+2y_2\hfill \\ y_2^{\prime }=\phantom{-}2y_1-\phantom{2}{y}_2\hfill \\ y_1\left(0\right)=3,y_2\left(0\right)=1\hfill \end{array}$

   2. $\begin{array}{c}{y}_1^{\prime }=\phantom{2}{y}_1-2y_2\hfill \\ y_2^{\prime }=2y_1+\phantom{2}{y}_2\hfill \\ y_1\left(0\right)=1,y_2\left(0\right)=-2\hfill \end{array}$

   3. $\begin{array}{c}{y}_1^{\prime }=2y_1-6y_3\hfill \\ y_2^{\prime }=y_1-3y_3\hfill \\ y_3^{\prime }=y_2-2y_3\hfill \\ y_1\left(0\right)=y_2\left(0\right)=y_3\left(0\right)=2\end{array}$

   4. $\begin{array}{c}{y}_1^{\prime }=y_1+2y_3\hfill \\ y_2^{\prime }=y_2-y_3\hfill \\ y_3^{\prime }=y_1+y_2+y_3\hfill \\ y_1\left(0\right)=y_2\left(0\right)=1,y_3\left(0\right)=4\end{array}$

3. Given

   $$\mathbf{\text{Y}}=c_1{e}^{{\lambda }_{1}t}{\mathbf{\text{x}}}_1+c_2{e}^{{\lambda }_{2}t}{\mathbf{\text{x}}}_2+\dots +c_n{e}^{{\lambda }_{n}t}{\mathbf{\text{x}}}_n$$

   is the solution to the initial value problem:

   $$\begin{array}{cc}{\mathbf{\text{Y}}}^{\prime }=A\mathbf{\text{Y}},& \mathbf{\text{Y}}\left(0\right)={\mathbf{\text{Y}}}_0\end{array}$$

   1. show that

      $${\mathbf{\text{Y}}}_0=c_1{\mathbf{\text{x}}}_1+c_2{\mathbf{\text{x}}}_2+\dots +c_n{\mathbf{\text{x}}}_n$$

   2. let $X=({\mathbf{\text{x}}}_1,\dots ,{\mathbf{\text{x}}}_n)$ and $\mathbf{\text{c}}=(c_1,\dots ,c_n{)}^T$. Assuming that the vectors ${\mathbf{\text{x}}}_1,\dots ,{\mathbf{\text{x}}}_n$ are linearly independent, show that $\mathbf{\text{c}}=X^{-1}{\mathbf{\text{Y}}}_0$.

4. Two tanks each contain 100 liters of a mixture. ...