
# 4.6E: Exercises

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In Exercises $$(4.6E.1)$$ to $$(4.6E.16)$$, find the general solution.

## Exercise $$\PageIndex{1}$$

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ {-1} & 2 \\ {-5} & 5 \end{array} \right] {\bf y} }$$

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## Exercise $$\PageIndex{2}$$

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ {-11} & 4 \\ {-26} & 9 \end{array} \right] {\bf y} }$$

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## Exercise $$\PageIndex{3}$$

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ 1 & 2 \\ {-4} & 5 \end{array} \right] {\bf y}}$$

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## Exercise $$\PageIndex{4}$$

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ 5 & {-6} \\ 3 & {-1} \end{array} \right] {\bf y}}$$

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## Exercise $$\PageIndex{5}$$

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ 3 & {-3} & 1 \\ 0 & 2 & 2 \\ 5 & 1 & 1 \end{array} \right] {\bf y}}$$

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## Exercise $$\PageIndex{6}$$

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ {-3} & 3 & 1 \\ 1 & {-5} & {-3} \\ {-3} & 7 & 3 \end{array} \right] {\bf y}}$$

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## Exercise $$\PageIndex{7}$$

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ 2 & 1 & {-1} \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{array} \right] {\bf y}}$$

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## Exercise $$\PageIndex{8}$$

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ {-3} & 1 & {-3} \\ 4 & {-1} & 2 \\ 4 & {-2} & 3 \end{array} \right] {\bf y}}$$

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## Exercise $$\PageIndex{9}$$

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ 5 & {-4} \\ {10} & 1 \end{array} \right] {\bf y}}$$

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## Exercise $$\PageIndex{10}$$

$$\displaystyle{{\bf y}' = {1\over3} \left[ \begin{array} \\ 7 & {-5} \\ 2 & 5 \end{array} \right] {\bf y}}$$

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## Exercise $$\PageIndex{11}$$

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ 3 & 2 \\ {-5} & 1 \end{array} \right] {\bf y}}$$

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## Exercise $$\PageIndex{12}$$

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ {34} & {52} \\ {-20} & {-30} \end{array} \right] {\bf y}}$$

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## Exercise $$\PageIndex{13}$$

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ 1 & 1 & 2 \\ 1 & 0 & {-1} \\ {-1} & {-2} & {-1} \end{array} \right] {\bf y}}$$

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## Exercise $$\PageIndex{14}$$

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ 3 & {-4} & {-2} \\ {-5} & 7 & {-8} \\ {-10} & {13} & {-8} \end{array} \right] {\bf y}}$$

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## Exercise $$\PageIndex{15}$$

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ 6 & 0 & {-3} \\ {-3} & 3 & 3 \\ 1 & {-2} & 6 \end{array} \right] {\bf y}'}$$

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## Exercise $$\PageIndex{16}$$

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ 1 & 2 & {-2} \\ 0 & 2 & {-1} \\ 1 & 0 & 0 \end{array} \right] {\bf y}'}$$

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In Exercises $$(4.6E.17)$$ to $$(4.6E.24)$$, solve the initial value problem.

## Exercise $$\PageIndex{17}$$

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ 4 & {-6} \\ 3 & {-2} \end{array} \right] {\bf y}, \quad {\bf y}(0) = \left[ \begin{array} \\ 5 \\ 2 \end{array} \right] }$$

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## Exercise $$\PageIndex{18}$$

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ 7 & {15} \\ {-3} & 1 \end{array} \right] {\bf y}, \quad {\bf y}(0) = \left[ \begin{array} \\ 5 \\ 1 \end{array} \right] }$$

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## Exercise $$\PageIndex{19}$$

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ 7 & {-15} \\ 3 & {-5} \end{array} \right] {\bf y}, \quad {\bf y}(0) = \left[ \begin{array} \\ {17} \\ 7 \end{array} \right]}$$

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## Exercise $$\PageIndex{20}$$

$$\displaystyle{{\bf y}' = {1\over 6} \left[ \begin{array} \\ 4 & {-2} \\ 5 & 2 \end{array} \right] {\bf y}, \quad {\bf y}(0) = \left[ \begin{array} \\ 1 \\ {-1} \end{array} \right] }$$

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## Exercise $$\PageIndex{21}$$

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ 5 & 2 & {-1} \\ {-3} & 2 & 2 \\ 1 & 3 & 2 \end{array} \right] {\bf y}, \quad {\bf y}(0) = \left[ \begin{array} \\ 4 \\ 0 \\ 6 \end{array} \right]}$$

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## Exercise $$\PageIndex{22}$$

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ 4 & 4 & 0 \\ 8 & {10} & {-20} \\ 2 & 3 & {-2} \end{array} \right] {\bf y}, \quad {\bf y}(0) = \left[ \begin{array} \\ 8 \\ 6 \\ 5 \end{array} \right] }$$

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## Exercise $$\PageIndex{23}$$

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ 1 & {15} & {-15} \\ {-6} & {18} & {-22} \\ {-3} & {11} & {-15} \end{array} \right] {\bf y}, \quad {\bf y}(0) = \left[ \begin{array} \\ {15} \\ {17} \\ {10} \end{array} \right]}$$

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## Exercise $$\PageIndex{24}$$

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ 4 & {-4} & 4 \\ {-10} & 3 & {15} \\ 2 & {-3} & 1 \end{array} \right] {\bf y}, \quad {\bf y}(0) = \left[ \begin{array} \\ {16} \\ {14} \\ 6 \end{array} \right] }$$

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## Exercise $$\PageIndex{25}$$

Suppose an $$n\times n$$ matrix $$A$$ with real entries has a complex eigenvalue $$\lambda=\alpha+i\beta$$ ($$\beta\ne0$$) with associated eigenvector $${\bf x}={\bf u}+i{\bf v}$$, where $${\bf u}$$ and $${\bf v}$$ have real components. Show that $${\bf u}$$ and $${\bf v}$$ are both nonzero.

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## Exercise $$\PageIndex{26}$$

Verify that

\begin{eqnarray*}
{\bf y}_1 = e^{\alpha t} ({\bf u} \cos \beta t - {\bf v} \sin \beta t) \quad \mbox{and} \quad {\bf y}_2 = e^{\alpha t} ({\bf u} \sin \beta t + {\bf v} \cos \beta t),
\end{eqnarray*}

are the real and imaginary parts of

\begin{eqnarray*}
e^{\alpha t} (\cos \beta t + i \sin \beta t) ({\bf u} + i {\bf v}).
\end{eqnarray*}

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## Exercise $$\PageIndex{27}$$

Show that if the vectors $${\bf u}$$ and $${\bf v}$$ are not both $${\bf 0}$$ and $$\beta\ne0$$ then the vector functions

\begin{eqnarray*}
{\bf y}_1 = e^{\alpha t}({\bf u} \cos \beta t - {\bf v} \sin \beta t) \quad \mbox{and} \quad {\bf y}_2 = e^{\alpha t} ({\bf u} \sin \beta t + {\bf v} \cos \beta t)
\end{eqnarray*}

are linearly independent on every interval.

Hint: There are two cases to consider: $$\{{\bf u},{\bf v}\}$$ linearly independent, and $$\{{\bf u},{\bf v}\}$$ linearly dependent. In either case, exploit the the linear independence of $$\{\cos\beta t,\sin\beta t\}$$ on every interval.

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## Exercise $$\PageIndex{28}$$

Suppose $${\bf u}=\displaystyle{ \left[ \begin{array} \\ {u_1} \\ {u_2} \end{array} \right] }$$ and $${\bf v}=\displaystyle{ \left[ \begin{array} \\ {v_1} \\ {v_2} \end{array} \right]}$$ are not orthogonal; that is, $$({\bf u},{\bf v})\ne0$$.

(a) Show that the quadratic equation

\begin{eqnarray*}
({\bf u}, {\bf v}) k^2 + (\| {\bf v} \|^2 - \| {\bf u} \|^2) k - ({\bf u}, {\bf v}) = 0
\end{eqnarray*}

has a positive root $$k_1$$ and a negative root $$k_2=-1/k_1$$.

(b) Let $${\bf u}_1^{(1)}={\bf u}-k_1{\bf v}$$, $${\bf v}_1^{(1)}={\bf v}+k_1{\bf u}$$, $${\bf u}_1^{(2)}={\bf u}-k_2{\bf v}$$, and $${\bf v}_1^{(2)}={\bf v}+k_2{\bf u}$$, so that $$({\bf u}_1^{(1)},{\bf v}_1^{(1)}) =({\bf u}_1^{(2)},{\bf v}_1^{(2)})=0$$, from the discussion given above. Show that

\begin{eqnarray*}
{\bf u}_1^{(2)} = {{\bf v}_1^{(1)} \over k_1} \quad \mbox{and} \quad {\bf v}_1^{(2)} = -{{\bf u}_1^{(1)} \over k_1}.
\end{eqnarray*}

(c) Let $${\bf U}_1$$, $${\bf V}_1$$, $${\bf U}_2$$, and $${\bf V}_2$$ be unit vectors in the directions of $${\bf u}_1^{(1)}$$, $${\bf v}_1^{(1)}$$, $${\bf u}_1^{(2)}$$, and $${\bf v}_1^{(2)}$$, respectively. Conclude from part (a) that $${\bf U}_2={\bf V}_1$$ and $${\bf V}_2=-{\bf U}_1$$, and that therefore the counterclockwise angles from $${\bf U}_1$$ to $${\bf V}_1$$ and from $${\bf U}_2$$ to $${\bf V}_2$$ are both $$\pi/2$$ or both $$-\pi/2$$.

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In Exercises $$(4.6E.29)$$ to $$(4.6E.32)$$, find vectors $${\bf U}$$ and $${\bf V}$$ parallel to the axes of symmetry of the trajectories, and plot some typical trajectories.

## Exercise $$\PageIndex{29}$$

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ 3 & {-5} \\ 5 & {-3} \end{array} \right]{\bf y}}$$

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## Exercise $$\PageIndex{30}$$

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ {-15} & {10} \\ {-25} & {15} \end{array} \right] {\bf y}}$$

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## Exercise $$\PageIndex{31}$$

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ {-4} & 8 \\ {-4} & 4 \end{array} \right] {\bf y}}$$

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## Exercise $$\PageIndex{32}$$

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ {-3} & {-15} \\ 3 & 3 \end{array} \right] {\bf y}}$$

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In Exercises $$(4.6E.33)$$ to $$(4.6E.40)$$, find vectors $${\bf U}$$ and $${\bf V}$$ parallel to the axes of symmetry of the shadow trajectories, and plot a typical trajectory.

## Exercise $$\PageIndex{33}$$

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ {-5} & 6 \\ {-12} & 7 \end{array} \right] {\bf y}}$$

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## Exercise $$\PageIndex{34}$$

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ 5 & {-12} \\ 6 & {-7} \end{array} \right] {\bf y}}$$

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## Exercise $$\PageIndex{35}$$

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ 4 & {-5} \\ 9 & {-2} \end{array} \right] {\bf y}}$$

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## Exercise $$\PageIndex{36}$$

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ {-4} & 9 \\ {-5} & 2 \end{array} \right] {\bf y}}$$

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## Exercise $$\PageIndex{37}$$

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ {-1} & {10} \\ {-10} & {-1} \end{array} \right] {\bf y}}$$

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## Exercise $$\PageIndex{38}$$

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ {-1} & {-5} \\ {20} & {-1} \end{array} \right] {\bf y}}$$

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## Exercise $$\PageIndex{39}$$

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ {-7} & {10} \\ {-10} & 9 \end{array} \right] {\bf y}}$$

## Exercise $$\PageIndex{40}$$
$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ {-7} & 6 \\ {-12} & 5 \end{array} \right] {\bf y}}$$