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Mathematics LibreTexts

4.2E: Exercises

  • Page ID
    18187
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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) \(\begin{array}{ccl}y'_1&=&2y_1 +\phantom{2}y_2 \\
    y_2'&=&\phantom{2}y_1 + 2y_2;\end{array} \quad
    {\bf y}=c_111e^{3t}+c_21{-1}e^t\)

    Answer

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

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

    Answer

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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*}

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

    Answer

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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*}

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

    Answer

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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*}

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

    Answer

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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 \((a,b)\).

    Answer

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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*}

    Answer

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

    Answer

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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.

    Answer

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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.

    Answer

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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.

    Answer

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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.

    Answer

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    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 \((a,b)\).

    Answer

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