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

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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

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

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.