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Mathematics LibreTexts

10.3E: Basic Theory of Homogeneous Linear Systems (Exercises)

  • Page ID
    18299
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    [exer:10.3.1] Prove: If \({\bf y}_1\), \({\bf y}_2\), …, \({\bf y}_n\) are solutions of \({\bf y}'=A(t){\bf y}\) on \((a,b)\), then any linear combination of \({\bf y}_1\), \({\bf y}_2\), …, \({\bf y}_n\) is also a solution of \({\bf y}'=A(t){\bf y}\) on \((a,b)\).

    [exer:10.3.2] In Section 5.1 the Wronskian of two solutions \(y_1\) and \(y_2\) of the scalar second order equation

    \[P_0(x)y''+P_1(x)y'+P_2(x)y=0 \eqno{\rm (A)}\]

    was defined to be

    \[W=\left|\begin{array}{cc} y_1&y_2 \\ y'_1&y'_2\end{array}\right|.\]

    Rewrite (A) as a system of first order equations and show that \(W\) is the Wronskian (as defined in this section) of two solutions of this system.

    Apply Eqn. Equation \ref{eq:10.3.6} to the system derived in

    a

    , and show that

    \[W(x)=W(x_0)\exp\left\{-\int^x_{x_0}{P_1(s)\over P_0(s)}\, ds\right\},\]

    which is the form of Abel’s formula given in Theorem 9.1.3.

    [exer:10.3.3] In Section 9.1 the Wronskian of \(n\) solutions \(y_1\), \(y_2\), …, \(y_n\) of the \(n-\)th order equation

    \[P_0(x)y^{(n)}+P_1(x)y^{(n-1)}+\cdots+P_n(x)y=0 \eqno{\rm (A)}\]

    was defined to be

    \[W=\left|\begin{array}{cccc} y_1&y_2&\cdots&y_n \

    \[5pt] y'_1&y'_2&\cdots&y_n'\

    \[5pt] \vdots&\vdots&\ddots&\vdots\

    \[5pt] y_1^{(n-1)}&y_2^{(n-1)}&\cdots&y_n^{(n-1)} \end{array}\right|.\]

    Rewrite (A) as a system of first order equations and show that \(W\) is the Wronskian (as defined in this section) of \(n\) solutions of this system.

    Apply Eqn. Equation \ref{eq:10.3.6} to the system derived in

    a

    , and show that

    \[W(x)=W(x_0)\exp\left\{-\int^x_{x_0}{P_1(s)\over P_0(s)}\, ds\right\},\]

    which is the form of Abel’s formula given in Theorem 9.1.3.

    [exer:10.3.4] 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 \(2\times 2\) system \({\bf y}'=A{\bf y}\) on \((a,b)\), and let

    \[Y=\twobytwo {y_{11}} {y_{12}} {y_{21}} {y_{22}}\mbox{\quad and \quad} W=\left|\begin{array}{cc} y_{11}&y_{12}\\y_{21}&y_{22}\end{array}\right|;\]

    thus, \(W\) is the Wronskian of \(\{{\bf y}_1,{\bf y}_2\}\).

    Deduce from the definition of determinant that

    \[W'=\left|\begin{array}{cc} {y'_{11}}&{y'_{12}}\\ {y_{21}}& {y_{22}}\end{array}\right| +\left|\begin{array}{cc} {y_{11}}&{y_{12}}\\ {y'_{21}}&{y'_{22}}\end{array}\right|.\]

    Use the equation \(Y'=A(t)Y\) and the definition of matrix multiplication to show that

    \[[y'_{11}\quad y'_{12}]=a_{11} [y_{11}\quad y_{12}]+a_{12} [y_{21} \quad y_{22}]\]

    and

    \[[y'_{21}\quad y'_{22}]=a_{21} [y_{11}\quad y_{12}]+a_{22} [y_{21}\quad y_{22}].\]

    Use properties of determinants to deduce from

    a

    and

    a

    that

    \[\left|\begin{array}{cc} {y'_{11}}&{y'_{12}}\\ {y_{21}}& {y_{22}}\end{array}\right|=a_{11}W\mbox{\quad and \quad} \left|\begin{array}{cc} {y_{11}}&{y_{12}}\\ {y'_{21}}&{y'_{22}}\end{array}\right|=a_{22}W.\]

    Conclude from

    c

    that

    \[W'=(a_{11}+a_{22})W,\]

    and use this to show that if \(a<t_0<b\) then

    \[W(t)=W(t_0)\exp\left(\int^t_{t_0} \left[a_{11}(s)+a_{22} (s) \right]\, ds\right)\quad a<t<b.\]

    [exer:10.3.5] Suppose the \(n\times n\) matrix \(A=A(t)\) is continuous on \((a,b)\). Let

    \[Y= \left[\begin{array}{cccc} y_{11}&y_{12}&\cdots&y_{1n} \\ y_{21}&y_{22}&\cdots&y_{2n} \\ \vdots&\vdots&\ddots&\vdots \\ y_{n1}&y_{n2}&\cdots&y_{nn} \end{array}\right],\]

    where the columns of \(Y\) are solutions of \({\bf y}'=A(t){\bf y}\). Let

    \[r_i=[y_{i1}\, y_{i2}\, \dots\, y_{in}]\]

    be the \(i\)th row of \(Y\), and let \(W\) be the determinant of \(Y\).

    Deduce from the definition of determinant that

    \[W'=W_1+W_2+\cdots+W_n,\]

    where, for \(1 \le m \le n\), the \(i\)th row of \(W_m\) is \(r_i\) if \(i \ne m\), and \(r'_m\) if \(i=m\).

    Use the equation \(Y'=A Y\) and the definition of matrix multiplication to show that

    \[r'_m=a_{m1}r_1+a_{m2} r_2+\cdots+a_{mn}r_n.\]

    Use properties of determinants to deduce from

    b

    that

    \[\det (W_m)=a_{mm}W.\]

    Conclude from

    a

    and

    c

    that

    \[W'=(a_{11}+a_{22}+\cdots+a_{nn})W,\]

    and use this to show that if \(a<t_0<b\) then

    \[W(t)=W(t_0)\exp\left( \int^t_{t_0}\big[a_{11}(s)+a_{22}(s)+\cdots+a_{nn}(s)]\, ds\right), \quad a < t < b.\]

    [exer:10.3.6] Suppose the \(n\times n\) matrix \(A\) is continuous on \((a,b)\) and \(t_0\) is a point in \((a,b)\). Let \(Y\) be a fundamental matrix for \({\bf y}'=A(t){\bf y}\) on \((a,b)\).

    Show that \(Y(t_0)\) is invertible.

    Show that if \({\bf k}\) is an arbitrary \(n\)-vector then the solution of the initial value problem

    \[{\bf y}'=A(t){\bf y},\quad {\bf y}(t_0)={\bf k}\]

    is

    \[{\bf y}=Y(t)Y^{-1}(t_0){\bf k}.\]

    [exer:10.3.7] Let

    \[A=\twobytwo2442, \quad {\bf y}_1=\left[\begin{array}{c} e^{6t} \\ e^{6t} \end{array}\right], \quad {\bf y}_2=\left[\begin{array}{r} e^{-2t} \\ -e^{-2t}\end{array}\right], \quad {\bf k}=\left[\begin{array}{r}-3 \\ 9\end{array}\right].\]

    Verify that \(\{{\bf y}_1,{\bf y}_2\}\) is a fundamental set of solutions for \({\bf y}'=A{\bf y}\).

    Solve the initial value problem

    \[{\bf y}'=A{\bf y},\quad {\bf y}(0)={\bf k}. \eqno{\rm(A)}\]

    Use the result of Exercise [exer:10.3.6}

    b

    to find a formula for the solution of (A) for an arbitrary initial vector \({\bf k}\).

    [exer:10.3.8] Repeat Exercise [exer:10.3.7} with

    \[A=\twobytwo {-2} {-2} {-5}1, \quad {\bf y}_1=\left[\begin{array}{r} e^{-4t} \\ e^{-4t}\end{array}\right], \quad {\bf y}_2=\left[ \begin{array}{r}-2e^{3t} \\ 5e^{3t}\end{array}\right], \quad {\bf k}=\left[\begin{array}{r} 10 \\-4\end{array}\right].\]

    [exer:10.3.9] Repeat Exercise [exer:10.3.7} with

    \[A=\twobytwo{-4} {-10} 3 7, \quad {\bf y}_1=\left[\begin{array}{r}-5e^{2t} \\ 3e^{2t} \end{array}\right], \quad {\bf y}_2=\left[\begin{array}{r} 2e^t \\-e^t \end{array}\right], \quad {\bf k}=\left[\begin{array}{r}-19 \\ 11\end{array} \right ].\]

    [exer:10.3.10] Repeat Exercise [exer:10.3.7} with

    \[A=\twobytwo 2 1 1 2, \quad {\bf y}_1=\left[\begin{array}{r} e^{3t} \\ e^{3t} \end{array}\right], \quad {\bf y}_2=\left[\begin{array}{r}e^t \\ -e^t\end{array}\right], \quad {\bf k}=\left[\begin{array}{r} 2 \\ 8 \end{array}\right].\]

    [exer:10.3.11] Let

    \[\begin{aligned} A&=&\threebythree 3 {-1} {-1} {-2} 3 24 {-1} {-2}, \\ {\bf y}_1&=&\left[\begin{array}{c} e^{2t} \\ 0 \\ e^{2t}\end{array} \right], \quad {\bf y}_2=\left[\begin{array}{c} e^{3t} \\-e^{3t} \\ e^{3t}\end{array}\right], \quad {\bf y}_3=\left[\begin{array}{c} e^{-t} \\-3e^{-t} \\ 7e^{-t} \end{array}\right], \quad {\bf k}=\left[\begin{array}{r} 2 \\-7 \\ 20\end{array}\right].\end{aligned}\]

    Verify that \(\{{\bf y}_1,{\bf y}_2,{\bf y}_3\}\) is a fundamental set of solutions for \({\bf y}'=A{\bf y}\).

    Solve the initial value problem

    \[{\bf y}'=A{\bf y}, \quad {\bf y}(0)={\bf k}. \eqno{\rm(A)}\]

    Use the result of Exercise [exer:10.3.6}

    b

    to find a formula for the solution of (A) for an arbitrary initial vector \({\bf k}\).

    [exer:10.3.12] Repeat Exercise [exer:10.3.11} with

    \[\begin{aligned} A&=&\threebythree 0 2 2 2 0 2 2 2 0, \\ {\bf y}_1&=&\left[\begin{array}{c}-e^{-2t} \\ 0 \\ e^{-2t} \end{array}\right], \quad {\bf y}_2=\left[\begin{array}{c}-e^{-2t} \\ e^{-2t} \\ 0\end{array}\right], \quad {\bf y}_3=\left[\begin{array}{c} e^{4t} \\ e^{4t} \\ e^{4t}\end{array} \right], \quad {\bf k}=\left[\begin{array}{r} 0 \\-9 \\ 12\end{array} \right].\end{aligned}\]

    [exer:10.3.13] Repeat Exercise [exer:10.3.11} with

    \[\begin{aligned} A&=&\threebythree {-1} 2 3 0 1 6 0 0 {-2}, \\ {\bf y}_1&=&\left[\begin{array}{c} e^t \\ e^t \\ 0\end{array}\right], \quad {\bf y}_2=\left[\begin{array}{c} e^{-t} \\ 0 \\ 0\end{array}\right], \quad {\bf y}_3=\left[\begin{array}{c} e^{-2t} \\-2e^{-2t} \\ e^{-2t}\end{array}\right], \quad {\bf k}=\left[\begin{array}{r} 5 \\ 5 \\-1 \end{array}\right].\end{aligned}\]

    [exer:10.3.14] Suppose \(Y\) and \(Z\) are fundamental matrices for the \(n\times n\) system \({\bf y}'=A(t){\bf y}\). Then some of the four matrices \(YZ^{-1}\), \(Y^{-1}Z\), \(Z^{-1}Y\), \(Z Y^{-1}\) are necessarily constant. Identify them and prove that they are constant.

    [exer:10.3.15] Suppose the columns of an \(n\times n\) matrix \(Y\) are solutions of the \(n\times n\) system \({\bf y}'=A{\bf y}\) and \(C\) is an \(n \times n\) constant matrix.

    Show that the matrix \(Z=YC\) satisfies the differential equation \(Z'=AZ\).

    Show that \(Z\) is a fundamental matrix for \({\bf y}'=A(t){\bf y}\) if and only if \(C\) is invertible and \(Y\) is a fundamental matrix for \({\bf y}'=A(t){\bf y}\).

    [exer:10.3.16] Suppose the \(n\times n\) matrix \(A=A(t)\) is continuous on \((a,b)\) and \(t_0\) is in \((a,b)\). For \(i=1\), \(2\), …, \(n\), let \({\bf y}_i\) be the solution of the initial value problem \({\bf y}_i'=A(t){\bf y}_i,\; {\bf y}_i(t_0)={\bf e}_i\), where

    \[{\bf e}_1=\left[\begin{array}{c} 1\\0\\ \vdots\\0\end{array}\right],\quad {\bf e}_2=\left[\begin{array}{c} 0\\1\\ \vdots\\0\end{array}\right],\quad\cdots\quad {\bf e}_n=\left[\begin{array}{c} 0\\0\\ \vdots\\1\end{array}\right];\]

    that is, the \(j\)th component of \({\bf e}_i\) is \(1\) if \(j=i\), or \(0\) if \(j\ne i\).

    Show that\(\{{\bf y}_1,{\bf y}_2,\dots,{\bf y}_n\}\) is a fundamental set of solutions of \({\bf y}'=A(t){\bf y}\) on \((a,b)\).

    Conclude from

    a

    and Exercise [exer:10.3.15} that \({\bf y}'= A(t){\bf y}\) has infinitely many fundamental sets of solutions on \((a,b)\).

    [exer:10.3.17] Show that \(Y\) is a fundamental matrix for the system \({\bf y}'=A(t){\bf y}\) if and only if \(Y^{-1}\) is a fundamental matrix for \({\bf y}'=- A^T(t){\bf y}\), where \(A^T\) denotes the transpose of \(A\).

    [exer:10.3.18] Let \(Z\) be the fundamental matrix for the constant coefficient system \({\bf y}'=A{\bf y}\) such that \(Z(0)=I\).

    Show that \(Z(t)Z(s)=Z(t+s)\) for all \(s\) and \(t\).

    Show that \((Z(t))^{-1}=Z(-t)\).

    The matrix \(Z\) defined above is sometimes denoted by \(e^{tA}\). Discuss the motivation for this notation.