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# 4.5E: Exercises

• • Contributed by William F. Trench
• Andrew G. Cowles Distinguished Professor Emeritus (Mathamatics) at Trinity University

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In Exercises $$(4.5E.1)$$ to $$(4.5E.12)$$, find the general solution.

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

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ 3 & 4 \\ {-1} & 7 \end{array} \right] {\bf y}}$$

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

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

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

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ {-7} & 4 \\ {-1} & {-11} \end{array} \right] {\bf y}}$$

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

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

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

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ 4 & {12} \\ {-3} & {-8} \end{array} \right] {\bf y}}$$

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

$$\displaystyle{{\bf y'} = \left[ \begin{array} \\ {-10} & 9 \\ {-4} & 2 \end{array} \right] {\bf y}}$$

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

$$\displaystyle{{\bf y'} = \left[ \begin{array} \\ {-13} & {16} \\ {-9} & {11} \end{array} \right] {\bf y}}$$

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

$$\displaystyle{{\bf y'} = \left[ \begin{array} \\ 0 & 2 & 1 \\ {-4} & 6 & 1 \\ 0 & 4 & 2 \end{array} \right] {\bf y}}$$

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

$$\displaystyle{{\bf y}' = {1\over3} \left[ \begin{array} \\ 1 & 1 & {-3} \\ {-4} & {-4} & 3 \\ {-2} & 1 & 0 \end{array} \right] {\bf y}}$$

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

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

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

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

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

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ 6 & {-5} & 3 \\ 2 & {-1} & 3 \\ 2 & 1 & 1 \end{array} \right] {\bf y}}$$

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In Exercises $$(4.5E.13)$$ to $$(4.5E.23)$$, solve the initial value problem.

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

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ {-11} & 8 \\ {-2} & {-3} \end{array} \right] {\bf y} , \quad {\bf y}(0) = \left[ \begin{array} \\ 6 \\ 2 \end{array} \right]}$$

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

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ {15} & {-9} \\ {16} & {-9} \end{array} \right] {\bf y} , \quad {\bf y}(0) = \left[ \begin{array} \\ 5 \\ 8 \end{array} \right] }$$

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

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ {-3} & {-4} \\ 1 & {-7} \end{array} \right] {\bf y} , \quad {\bf y}(0) = \left[ \begin{array} \\ 2 \\ 3 \end{array} \right]}$$

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

$${\bf y}' = \left[ \begin{array} \\ -7 & 24 \\ -6 & 17 \end{array} \right] {\bf y} , \quad {\bf y}(0) = \left[ \begin{array} \\ 3 \\ 1 \end{array} \right]$$

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

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ {-7} & 3 \\ {-3} & {-1} \end{array} \right] {\bf y} , \quad {\bf y}(0) = \left[ \begin{array} \\ 0 \\ 2 \end{array} \right]}$$

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

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ {-1} & 1 & 0 \\ 1 & {-1} & {-2} \\ {-1} & {-1} & {-1} \end{array} \right] {\bf y} , \quad {\bf y}(0) = \left[ \begin{array} \\ 6 \\ 5 \\ {-7} \end{array} \right] }$$

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

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

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

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ {-7} & {-4} & 4 \\ {-1} & 0 & 1 \\ {-9} & {-5} & 6 \end{array} \right] {\bf y} , \quad \bf {\bf y}(0) = \left[ \begin{array} \\ {-6} \\ 9 \\ {-1} \end{array} \right]}$$

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

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

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

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ 4 & {-8} & {-4} \\ {-3} & {-1} & {-3} \\ 1 & {-1} & 9 \end{array} \right] {\bf y} , \quad {\bf y}(0) = \left[ \begin{array} \\ {-4} \\ 1 \\ {-3} \end{array} \right]}$$

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

$$\displaystyle{{\bf y}'= \left[ \begin{array} \\ {-5} & {-1} & {11} \\ {-7} & 1 & {13} \\ {-4} & 0 & 8 \end{array} \right] {\bf y} , \quad {\bf y}(0) = \left[ \begin{array} \\ 0 \\ 2 \\ 2 \end{array} \right] }$$

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The coefficient matrices in Exercises $$(4.5E.24)$$ to $$(4.5E.32)$$ have eigenvalues of multiplicity $$3$$. Find the general solution.

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

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ 5 & {-1} & 1 \\ {-1} & 9 & {-3} \\ {-2} & 2 & 4 \end{array} \right] {\bf y}}$$

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

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ 1 & {10} & {-12} \\ 2 & 2 & 3 \\ 2 & {-1} & 6 \end{array} \right] {\bf y}}$$

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

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ {-6} & {-4} & {-4} \\ 2 & {-1} & 1 \\ 2 & 3 & 1 \end{array} \right] {\bf y}}$$

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

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

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

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ {-2} & {-12} & {10} \\ 2 & {-24} & {11} \\ 2 & {-24} & 8 \end{array} \right] {\bf y}}$$

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

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ {-1} & {-12} & 8 \\ 1 & {-9} & 4 \\ 1 & {-6} & 1 \end{array} \right] {\bf y}}$$

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

$$\displaystyle{{\bf y}' = \left[ \begin{array} \\ {-4} & 0 & {-1} \\ {-1} & {-3} & {-1} \\ 1 & 0 & {-2} \end{array} \right] {\bf y}}$$

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

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

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

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

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

Under the assumptions of Theorem $$(4.5.1)$$, suppose $${\bf u}$$ and $$\hat{\bf u}$$ are vectors such that

\begin{eqnarray*}
(A - \lambda_1I) {\bf u} = {\bf x} \quad \mbox{and} \quad (A - \lambda_1I) \hat {\bf u} = {\bf x},
\end{eqnarray*}

and let

\begin{eqnarray*}
{\bf y}_2 = {\bf u} e^{\lambda_1t} + {\bf x} t e^{\lambda_1t} \quad \mbox{and} \quad \hat {\bf y}_2 = \hat {\bf u} e^{\lambda_1t} + {\bf x} te^{\lambda_1t}.
\end{eqnarray*}

Show that $${\bf y}_2-\hat{\bf y}_2$$ is a scalar multiple of $${\bf y}_1={\bf x}e^{\lambda_1t}$$.

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

Under the assumptions of Theorem $$(4.5.2)$$, let

\begin{eqnarray*}
{\bf y}_1 &=&{\bf x} e^{\lambda_1t},\\
{\bf y}_2&=&{\bf u}e^{\lambda_1t}+{\bf x} te^{\lambda_1t},\mbox{
and }\\
{\bf y}_3&=&{\bf v}e^{\lambda_1t}+{\bf u}te^{\lambda_1t}+{\bf
x} {t^2e^{\lambda_1t}\over2}.
\end{eqnarray*}

Complete the proof of Theorem $$(4.5.2)$$ by showing that $${\bf y}_3$$ is a solution of $${\bf y}'=A{\bf y}$$ and that $$\{{\bf y}_1,{\bf y}_2,{\bf y}_3\}$$ is linearly independent.

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

Suppose the matrix

\begin{eqnarray*}
A = \left[ \begin{array} \\ a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right]
\end{eqnarray*}

has a repeated eigenvalue $$\lambda_1$$ and the associated eigenspace is one-dimensional. Let $${\bf x}$$ be a $$\lambda_1$$-eigenvector of $$A$$. Show that if $$(A-\lambda_1I){\bf u}_1={\bf x}$$ and $$(A-\lambda_1I){\bf u}_2={\bf x}$$, then $${\bf u}_2-{\bf u}_1$$ is parallel to $${\bf x}$$. Conclude from this that all vectors $${\bf u}$$ such that $$(A-\lambda_1I){\bf u}={\bf x}$$ define the same positive and negative half-planes with respect to the line $$L$$ through the origin parallel to $${\bf x}$$.

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In Exercises $$(4.5E.36)$$ to $$(4.5E.45)$$, plot trajectories of the given system.

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

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

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

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

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

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

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

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

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## Exercise $$\PageIndex{40}$$

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

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## Exercise $$\PageIndex{41}$$

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

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## Exercise $$\PageIndex{42}$$

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

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## Exercise $$\PageIndex{43}$$

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

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## Exercise $$\PageIndex{44}$$

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

## Exercise $$\PageIndex{45}$$
$${\bf y}' = \displaystyle{ \left[ \begin{array} \\ 0 & {-4} \\ 1 & {-4} \end{array} \right] }{\bf y}$$