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Mathematics LibreTexts

10.2E: Linear Systems of Differential Equations (Exercises)

  • Page ID
    18298
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    [exer:10.2.1] 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 \(c_2\).

    \(\begin{array}{ccl}y'_1&=&2y_1 + 4y_2\\ y_2'&=&4y_1+2y_2;\end{array} \quad {\bf y}=c_1\twocol11e^{6t}+c_2\twocol1{-1}e^{-2t}\)

    \(\begin{array}{ccl}y'_1&=&-2y_1 - 2y_2\\ y_2'&=&-5y_1 + \phantom{2}y_2;\end{array} \quad {\bf y}=c_1\twocol11e^{-4t}+c_2\twocol{-2}5e^{3t}\)

    \(\begin{array}{ccr}y'_1&=&-4y_1 -10y_2\\ y_2'&=&3y_1 + \phantom{1}7y_2;\end{array} \quad {\bf y}=c_1\twocol{-5}3e^{2t}+c_2\twocol2{-1}e^t\)

    \(\begin{array}{ccl}y'_1&=&2y_1 +\phantom{2}y_2 \\ y_2'&=&\phantom{2}y_1 + 2y_2;\end{array} \quad {\bf y}=c_1\twocol11e^{3t}+c_2\twocol1{-1}e^t\)

    [exer:10.2.2] 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 \(c_3\).

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

    \({\bf y}=c_1\threecol110e^t+c_2\threecol100e^{-t}+c_3\threecol1{-2}1e^{-2t}\)

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

    \({\bf y}=c_1\threecol{-1}01e^{-2t}+c_2\threecol0{-1}1e^{-2t}+c_3\threecol111e^{4t}\)

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

    \({\bf y}=c_1\threecol{-1}01e^{-3t}+c_2\threecol0{-1}1e^{-3t}+c_3\threecol111e^{3t}\)

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

    \({\bf y}=c_1\threecol101e^{2t}+c_2\threecol1{-1}1e^{3t}+c_3\threecol1{-3}7e^{-t}\)

    [exer:10.2.3] Rewrite the initial value problem in matrix form and verify that the given vector function is a solution.

    \(\begin{array}{ccl}y'_1 &=&\phantom{-2}y_1+\phantom{4}y_2\\ y_2'&=&-2y_1 + 4y_2,\end{array} \begin{array}{ccr}y_1(0)&=&1\\y_2(0)&=&0;\end{array}\) \({\bf y}=2\twocol11e^{2t}-\twocol12e^{3t}\)

    \(\begin{array}{ccl}y'_1 &=&5y_1 + 3y_2 \\ y_2'&=&- y_1 + y_2,\end{array} \begin{array}{ccr}y_1(0)&=&12\\y_2(0)&=&-6;\end{array}\) \({\bf y}=3\twocol1{-1}e^{2t}+3\twocol3{-1}e^{4t}\)

    [exer:10.2.4] Rewrite the initial value problem in matrix form and verify that the given vector function is a solution.

    \(\begin{array}{ccr}y'_1&=&6y_1 + 4y_2 + 4y_3 \\ y_2'&=&-7y_1 -2y_2 - y_3,\\y_3'&=&7y_1 + 4y_2 + 3y_3\end{array},\; \begin{array}{ccr}y_1(0)&=&3\\ y_2(0)&=&-6\\ y_3(0)&=&4\end{array}\)

    \({\bf y}=\threecol1{-1}1e^{6t}+2\threecol1{-2}1e^{2t}+\threecol0{-1}1e^{-t}\)

    \(\begin{array}{ccr}y'_1&=& \phantom{-}8y_1 + 7y_2 +\phantom{1}7y_3 \\ y_2'&=&-5y_1 -6y_2 -\phantom{1}9y_3,\\y_3'&=& \phantom{-}5y_1 + 7y_2 +10y_3,\end{array}\ \begin{array}{ccr}y_1(0)&=&2\\ y_2(0)&=&-4\\ y_3(0)&=&3\end{array}\)

    \({\bf y}=\threecol1{-1}1e^{8t}+\threecol0{-1}1e^{3t}+\threecol1{-2}1e^t\)

    [exer:10.2.5] 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 \(c_2\).

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

    \({\bf y}=c_1\left[\begin{array}{c}2\cos t\\3\cos t-\sin t\end{array}\right]+c_2\left[\begin{array}{c}2\sin t\\3\sin t+\cos t \end{array}\right]+\twocol1t\)

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

    \({\bf y}=c_1\twocol{-1}1e^{2t}+c_2\left[\begin{array}{c}1+t\\-t\end{array} \right]e^{2t}+\twocol13e^t\)

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

    \({\bf y}=c_1\twocol21e^{-3t}+c_2\twocol{-2}1e^t+\left[\begin{array}{c} e^{3t}\\2te^t\end{array}\right]\)

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

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

    [exer:10.2.6] Convert the linear scalar equation

    \[P_0(t)y^{(n)}+P_1(t)y^{(n-1)}+\cdots+P_n(t)y(t)=F(t) \eqno{\rm (A)}\]

    into an equivalent \(n\times n\) system

    \[{\bf y'}=A(t){\bf y}+{\bf f}(t),\]

    and show that \(A\) and \({\bf f}\) are continuous on an interval \((a,b)\) if and only if (A) is normal on \((a,b)\).

    [exer:10.2.7] A matrix function

    \[Q(t)=\matfunc qrs\]

    is said to be differentiable if its entries \(\{q_{ij}\}\) are differentiable. Then the derivative \(Q'\) is defined by

    \[Q'(t)=\matfunc {q'}rs.\]

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

    \[(c_1P+c_2Q)'=c_1P'+c_2Q'.\]

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

    \[(PQ)'=P'Q+PQ'.\]

    [exer:10.2.8] Verify that \(Y' = AY\).

    \(\{Y = \twobytwo {e^{6t}}{e^{-2t}} {e^{6t}}{-e^{-2t}}, \quad A = \twobytwo 2 4 4 2}\)

    \(\{Y = \twobytwo {e^{-4t}} {-2e^{3t}} {e^{-4t}} {5e^{3t}}, \quad A = \twobytwo {-2} {-2} {-5} {1}}\)

    \(\{Y = \twobytwo {-5e^{2t}} {2e^t} {3e^{2t}} {-e^t}, \quad A = \twobytwo {-4} {-10} 3 7}\)

    \(\{Y = \twobytwo {e^{3t}} {e^t} {e^{3t}} {-e^t}, \quad A = \twobytwo 2 1 1 2}\)

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

    \(\{Y = \cthreebythree {-e^{-2t}} {-e^{-2t}} {e^{4t}} 0 {\phantom{-} e^{-2t}} {e^{4t}} {e^{-2t}} 0 {e^{4t}}, \quad A = \threebythree 0 2 2 2 0 2 2 2 0}\)

    \(\{Y = \cthreebythree {e^{3t}} {e^{-3t}} 0 {e^{3t}} 0 {-e^{-3t}} {e^{3t}} {e^{-3t}} {\phantom{-}e^{-3t}}, \quad A = \threebythree {-9}66{-6}36{-6}63}\)

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

    [exer:10.2.9] Suppose

    \[{\bf y}_1=\twocol{y_{11}}{y_{21}}\mbox{\quad and \quad}{\bf y}_2=\twocol{y_{12}}{y_{22}}\]

    are solutions of the homogeneous system

    \[{\bf y}'=A(t){\bf y}, \eqno{\rm (A)}\]

    and define

    \[Y= \twobytwo{y_{11}}{y_{12}}{y_{21}}{y_{22}}.\]

    Show that \(Y'=AY\).

    Show that if \({\bf c}\) is a constant vector then \({\bf y}= Y{\bf c}\) is a solution of (A).

    State generalizations of

    a

    and

    b

    for \(n\times n\) systems.

    [exer:10.2.10] Suppose \(Y\) is a differentiable square matrix.

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

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

    State how the results obtained in

    a

    and

    b

    are analogous to results from calculus concerning scalar functions.

    [exer:10.2.11] It can be shown that if \(Y\) is a differentiable and invertible square matrix function, then \(Y^{-1}\) is differentiable.

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

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

    State how the results obtained in

    a

    and

    b

    are analogous to results from calculus concerning scalar functions.

    [exer:10.2.12] Show that Theorem [thmtype:10.2.1} implies Theorem [thmtype:9.1.1}.

    [exer:10.2.13] 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 \((a,b)\).