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# 4.2E: Exercises

• • Contributed by William F. Trench
• Andrew G. Cowles Distinguished Professor Emeritus (Mathamatics) at Trinity University
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## Exercise $$\PageIndex{1}$$

Rewrite the system in matrix form and verify that the given vector function satisfies the system for any choice of the constants $$c_1$$ and $$c_2$$.

(a) $$\begin{array}{ccl}y'_1&=&2y_1 + 4y_2\\ y_2'&=&4y_1+2y_2;\end{array} \quad {\bf y}=c_111e^{6t}+c_21{-1}e^{-2t}$$

(b) $$\begin{array}{ccl}y'_1&=&-2y_1 - 2y_2\\ y_2'&=&-5y_1 + \phantom{2}y_2;\end{array} \quad {\bf y}=c_111e^{-4t}+c_2{-2}5e^{3t}$$

(c) $$\begin{array}{ccr}y'_1&=&-4y_1 -10y_2\\ y_2'&=&3y_1 + \phantom{1}7y_2;\end{array} \quad {\bf y}=c_1{-5}3e^{2t}+c_2 2{-1}e^t$$

(d) $$\begin{array}{ccl}y'_1&=&2y_1 +\phantom{2}y_2 \\ y_2'&=&\phantom{2}y_1 + 2y_2;\end{array} \quad {\bf y}=c_111e^{3t}+c_21{-1}e^t$$

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

Rewrite the system in matrix form and verify that the given vector function satisfies the system for any choice of the constants $$c_1$$, $$c_2$$, and $$c_3$$.

(a) $$\begin{array}{ccr}y'_1&=&- y_1+2y_2 + 3y_3 \\ y_2'&=&y_2 + 6y_3\\y_3'&=&- 2y_3;\end{array}$$

$${\bf y}=c_1110e^t+c_2100e^{-t}+c_31{-2}1e^{-2t}$$

(b) $$\begin{array}{ccc}y'_1&=&\phantom{2y_1+}2y_2 + 2y_3 \\ y_2'&=&2y_1\phantom{+2y_2} + 2y_3\\y_3'&=&2y_1 + 2y_2;\phantom{+2y_3}\end{array}$$

$${\bf y}=c_1{-1}01e^{-2t}+c_20{-1}1e^{-2t}+c_3111e^{4t}$$

(c) $$\begin{array}{ccr}y'_1&=&-y_1 +2y_2 + 2y_3\\ y_2'&=&2y_1 -\phantom{2}y_2 +2y_3\\y_3'&=&2y_1 + 2y_2 -\phantom{2}y_3;\end{array}$$

$${\bf y}=c_1{-1}01e^{-3t}+c_20{-1}1e^{-3t}+c_3111e^{3t}$$

(d) $$\begin{array}{ccr}y'_1&=&3y_1 - \phantom{2}y_2 -\phantom{2}y_3 \\ y_2'&=&-2y_1 + 3y_2 + 2y_3\\y_3'&=&\phantom{-}4y_1 -\phantom{3}y_2 - 2y_3;\end{array}$$

$${\bf y}=c_1101e^{2t}+c_21{-1}1e^{3t}+c_31{-3}7e^{-t}$$

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

Rewrite the initial value problem in matrix form and verify that the given vector function is a solution.

(a) \begin{eqnarray*}y'_1 &=&\phantom{-2}y_1+\phantom{4}y_2\\
y_2'&=&-2y_1 + 4y_2,\end{eqnarray*}

\begin{eqnarray*}y_1(0)&=&1\\y_2(0)&=&0;\end{eqnarray*}

$${\bf y}=211e^{2t}-12e^{3t}$$

(b) \begin{eqnarray*}y'_1 &=&5y_1 + 3y_2 \\
y_2'&=&- y_1 + y_2,\end{eqnarray*}

\begin{eqnarray*}y_1(0)&=&12\\y_2(0)&=&-6;\end{eqnarray*}

$${\bf y}=31{-1}e^{2t}+33{-1}e^{4t}$$

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

Rewrite the initial value problem in matrix form and verify that the given vector function is a solution.

(a) \begin{eqnarray*}y'_1&=&6y_1 + 4y_2 + 4y_3 \\
y_2'&=&-7y_1 -2y_2 - y_3,\\y_3'&=&7y_1 + 4y_2 + 3y_3,\end{eqnarray*}

\begin{eqnarray*}y_1(0)&=&3\\ y_2(0)&=&-6\\ y_3(0)&=&4\end{eqnarray*}

$${\bf y}=1{-1}1e^{6t}+21{-2}1e^{2t}+0{-1}1e^{-t}$$

(b) \begin{eqnarray*}y'_1&=& \phantom{-}8y_1 + 7y_2 +\phantom{1}7y_3 \\
y_2'&=&-5y_1 -6y_2 -\phantom{1}9y_3,\\y_3'&=& \phantom{-}5y_1 + 7y_2 +10y_3,\end{eqnarray*}

\begin{eqnarray*}y_1(0)&=&2\\ y_2(0)&=&-4\\ y_3(0)&=&3\end{eqnarray*}

$${\bf y}=1{-1}1e^{8t}+0{-1}1e^{3t}+1{-2}1e^t$$

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

Rewrite the system in matrix form and verify that the given vector function satisfies the ystem for any choice of the constants $$c_1$$ and $$c_2$$.

(a) \begin{eqnarray*}y'_1&=&-3y_1+2y_2+3-2t \\ y_2'&=&-5y_1+3y_2+6-3t\end{eqnarray*}

$${\bf y}=c_1 \left[\begin{array}2\cos t\\3\cos t-\sin t\end{array} \right] + c_2 \left[ \begin{array}2\sin t\\3\sin t+\cos t \end{array} \right] + 1t$$

(b) \begin{eqnarray*}y'_1&=&3y_1+y_2-5e^t \\ y_2'&=&-y_1+y_2+e^t\end{eqnarray*}

$${\bf y}=c_1{-1}1e^{2t}+c_2\left[\begin{array}1+t\\-t\end{array} \right]e^{2t}+13e^t$$

(c) \begin{eqnarray*}y'_1&=&-y_1-4y_2+4e^t+8te^t \\ y_2'&=&-y_1-\phantom{4}y_2+e^{3t}+(4t+2)e^t\end{eqnarray*}

$${\bf y}=c_121e^{-3t}+c_2{-2}1e^t+\left[\begin{array}{c} e^{3t}\\2te^t\end{array}\right]$$

(d) \begin{eqnarray*}y'_1&=&-6y_1-3y_2+14e^{2t}+12e^t \\ y_2'&=&\phantom{6}y_1-2y_2+7e^{2t}-12e^t\end{eqnarray*}

$${\bf y}=c_1{-3}1e^{-5t}+c_2{-1}1e^{-3t}+ \left[\begin{array}{c}e^{2t}+3e^t\\2e^{2t}-3e^t\end{array}\right]$$

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

Convert the linear scalar equation

\begin{equation} \label{eq:4.2E.1}
P_0 (t) y^{(n)} + P_1 (t) y^{(n-1)} + \cdots + P_n (t) y(t) = F(t)
\end{equation}

into an equivalent $$n\times n$$ system

\begin{eqnarray*}
{\bf y'} = A(t) {\bf y} + {\bf f}(t),
\end{eqnarray*}

and show that $$A$$ and $${\bf f}$$ are continuous on an interval $$(a,b)$$ if and only if \eqref{eq:4.2E.1} is normal on $$(a,b)$$.

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

A matrix function

\begin{eqnarray*}
Q(t) = q_{rs}
\end{eqnarray*}

is said to be $$\textcolor{blue}{\mbox{differentiable}}$$ if its entries $$\{q_{ij}\}$$ are differentiable. Then the $$\textcolor{blue}{\mbox{derivative}}$$ $$Q'$$ is defined by

\begin{eqnarray*}
Q'(t) = q'_{rs}.
\end{eqnarray*}

(a) Prove: If $$P$$ and $$Q$$ are differentiable matrices such that $$P+Q$$ is defined and if $$c_1$$ and $$c_2$$ are constants, then

\begin{eqnarray*}
(c_1 P + c_2 Q)' = c_1 P' + c_2 Q'.
\end{eqnarray*}

(b) Prove: If $$P$$ and $$Q$$ are differentiable matrices such that $$PQ$$ is defined, then

\begin{eqnarray*}
(PQ)' = P'Q + PQ'.
\end{eqnarray*}

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

Verify that $$Y' = AY$$.

(a) $$Y = \left[ \begin{array} \\ {e^{6t}} & {e^{-2t}}\\ {e^{6t}} & {-e^{-2t}} \end{array} \right], \quad A = \left[ \begin{array} \\ 2 & 4 \\ 4 & 2 \end{array} \right]$$

(b) $$Y = \left[\begin{array} \\ {e^{-4t}} & {-2e^{3t}} \\ {e^{-4t}} & {5e^{3t}} \end{array} \right], \quad A = \left[ \begin{array} \\ {-2} & {-2} \\ {-5} & {1} \end{array} \right]$$

(c) $$Y = \left[ \begin{array} \\ {-5e^{2t}} & {2e^t} \\ {3e^{2t}} & {-e^t} \end{array} \right], \quad A = \left[ \begin{array} \\ {-4} & {-10} \\ 3 & 7 \end{array} \right]$$

(d) $$Y = \left[ \begin{array} \\ {e^{3t}} & {e^t} \\ {e^{3t}} & {-e^t} \end{array} \right], \quad A = \left[ \begin{array} \\ 2 & 1 \\ 1 & 2 \end{array} \right]$$

(e) $$Y = \left[ \begin{array} \\ {e^t} & {e^{-t}} & {e^{-2t}} \\ {e^t} & 0 & {-2e^{-2t}} \\ 0 & 0 & {e^{-2t}} \end{array} \right], \quad A = \left[ \begin{array} \\ {-1} & 2 & 3 \\ 0 & 1 & 6 \\ 0 & 0 & {-2} \end{array} \right]$$

(f) $$Y = \left[ \begin{array} \\ {-e^{-2t}} & {-e^{-2t}} & {e^{4t}} \\ 0 & {\phantom{-} e^{-2t}} & {e^{4t}} \\ {e^{-2t}} & 0 & { e^{4t}} \end{array} \right], \quad A = \left[ \begin{array} \\ 0 & 2 & 2 \\ 2 & 0 & 2 \\ 2 & 2 & 0 \end{array} \right]$$

(g) $$Y = \left[ \begin{array} \\ {e^{3t}} & {e^{-3t}} & 0 \\ {e^{3t}} & 0 & {-e{-3t}} \\ {e^{3t}} & {e^{-3t}} & {\phantom{-}e^{-3t}} \end{array} \right], \quad A = \left[ \begin{array} \\ {-9} & 6 & 6 \\ {-6} & 3 & 6 \\ {-6} & 6 & 3 \end{array} \right]$$

(h) $$Y = \left[ \begin{array} \\ {e^{2t}} & {e^{3t}} & {e^{-t}} \\ 0 & {-e^{-3t}} & {-3e^{-t}} \\ {e^{2t}} & {e^{3t}} & {7e^{-t}} \end{array} \right], \quad A = \left[ \begin{array} \\ 3 & {-1} & {-1} \\ {-2} & 3 & 2 \\ 4 & {-1} & {-2} \end{array} \right]$$

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

Suppose $${\bf y}_1 = \left[ \begin{array} \\ y_{11} \\ y_{21} \end{array} \right] \quad \mbox{and} \quad {\bf y}_2 = \left[ \begin{array} \\ y_{12} \\ y_{22} \end{array} \right]$$ are solutions of the homogeneous system

\begin{equation} \label{eq:4.2E.2}
{\bf y}' = A(t) {\bf y},
\end{equation}

and define $$Y = \left[ \begin{array} \\ \; y_{11} \; y_{12} \\ \; y_{21} \; y_{22} \end{array} \right]$$.

(a) Show that $$Y'=AY$$.

(b) Show that if $${\bf c}$$ is a constant vector then $${\bf y}= Y{\bf c}$$ is a solution of \eqref{eq:4.2E.2}.

(c) State generalizations of part (a) and part (b) for $$n\times n$$ systems.

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

Suppose $$Y$$ is a differentiable square matrix.

(a) Find a formula for the derivative of $$Y^2$$.

(b) Find a formula for the derivative of $$Y^n$$, where $$n$$ is any positive integer.

(c) State how the results obtained in part (a\) and part (b\) are analogous to results from calculus concerning scalar functions.

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

It can be shown that if $$Y$$ is a differentiable and invertible square matrix function, then $$Y^{-1}$$ is differentiable.

(a) Show that $$(Y^{-1})' = -Y^{-1}Y'Y^{-1}$$.
Hint: Differentiate the identity $$Y^{-1}Y=I$$.

(b) Find the derivative of $$Y^{-n}=\left(Y^{-1}\right)^n$$, where $$n$$ is a positive integer.

(c) State how the results obtained in part (a) and part (b) are analogous to results from calculus concerning scalar functions.

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

Show that Theorem $$(4.2.1)$$ implies Theorem $$(3.1.1)$$.

Hint: Write the scalar equation

\begin{eqnarray*}
P_0(x)y^{(n)} + P_1(x)y^{(n-1)} + \cdots + P_n(x)y = F(x)
\end{eqnarray*}

as an $$n\times n$$ system of linear equations.

## Exercise $$\PageIndex{13}$$
Suppose $${\bf y}$$ is a solution of the $$n\times n$$ system $${\bf y}'=A(t){\bf y}$$ on $$(a,b)$$, and that the $$n\times n$$ matrix $$P$$ is invertible and differentiable on $$(a,b)$$. Find a matrix $$B$$ such that the function $${\bf x}=P{\bf y}$$ is a solution of $${\bf x}'=B{\bf x}$$ on $$(a,b)$$.