
# 4.3E: Exercises

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### Exercise $$\PageIndex{1}$$

Prove: If $${\bf y}_1$$, $${\bf y}_2$$, $$\dots$$, $${\bf y}_n$$ are solutions of $${\bf y}'=A(t){\bf y}$$ on $$(a,b)$$, then any linear combination of $${\bf y}_1$$, $${\bf y}_2$$, $$\dots$$, $${\bf y}_n$$ is also a solution of $${\bf y}'=A(t){\bf y}$$ on $$(a,b)$$.

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

In Section 2.1 the Wronskian of two solutions $$y_1$$ and $$y_2$$ of the scalar second order equation

\label{eq:4.3E.1}
P_0(x)y'' + P_1(x)y' + P_2(x)y = 0

was defined to be

\begin{eqnarray*}
W = \left| \begin{array} \\ y_1 & y_2 \\ y'_1 & y'_2 \end{array} \right|.
\end{eqnarray*}

(a) Rewrite \eqref{eq:4.3E.1} 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.

(b) Apply Equation $$(4.3.6)$$ to the system derived in part (a), and show that

\begin{eqnarray*}
W(x) = W(x_0)\exp\left\{-\int^x_{x_0}{P_1(s)\over P_0(s)}\,ds\right\},
\end{eqnarray*}

which is the form of Abel's formula given in Theorem $$(3.1.3$$)

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

In Section 3.1, the Wronskian of $$n$$ solutions $$y_1$$, $$y_2$$, $$\dots$$, $$y_n$$ of the $$n$$th order equation

\label{eq:4.3E.2}
P_0(x)y^{(n)} + P_1(x)y^{(n-1)} + \cdots + P_n(x)y = 0

was defined to be

\begin{eqnarray*}
W = \left| \begin{array} \\ y_1 & y_2 & \cdots & y_n \\ y'_1 & y'_2 & \cdots & y'_n \\ \vdots & \vdots & \ddots & \vdots \\ y_1^{(n-1)} & y_2^{(n-1)} & \cdots & y_n^{(n-1)} \end{array} \right|.
\end{eqnarray*}

(a) Rewrite \eqref{eq:4.3E.2} 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.

(b) Apply Equation $$(4.3.6)$$ to the system  derived in part (a), and show that

\begin{eqnarray*}
W(x) = W(x_0)\exp\left\{-\int^x_{x_0}{P_1(s)\over P_0(s)}\,ds\right\},
\end{eqnarray*}

which is the form of Abel's formula given in Theorem $$(3.1.3$$).

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

Suppose

\begin{eqnarray*}
{\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]
\end{eqnarray*}

are solutions of the $$2\times 2$$ system $${\bf y}'=A{\bf y}$$ on $$(a,b)$$, and let

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

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

(a) Deduce from the definition of determinant that

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

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

\begin{eqnarray*}
\end{eqnarray*}

and

\begin{eqnarray*}
\end{eqnarray*}

(c) Use properties of determinants to deduce from part (a) and part (b) that

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

(d) Conclude from part (c) that

\begin{eqnarray*}
W' = (a_{11} + a_{22}) W,
\end{eqnarray*}

and use this to show that if $$a<t_0<b$$ then

\begin{eqnarray*}
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.
\end{eqnarray*}

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

Suppose the $$n\times n$$ matrix $$A=A(t)$$ is continuous on $$(a,b)$$. Let

\begin{eqnarray*}
Y = \left[\begin{array} \\ 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],
\end{eqnarray*}

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

\begin{eqnarray*}
r_i = [y_{i1} \, y_{i2} \, \dots \, y_{in}]
\end{eqnarray*}

be the $$i$$th row of $$Y$$, and let $$W$$ be the determinant of $$Y$$.

(a) Deduce from the definition of determinant that

\begin{eqnarray*}
W' = W_1 + W_2 + \cdots + W_n,
\end{eqnarray*}

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 $$i=m$$.

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

\begin{eqnarray*}
r'_m = a_{m1} r_1 + a_{m2} r_2 + \cdots + a_{mn} r_n.
\end{eqnarray*}

(c) Use properties of determinants to deduce from part (b) that

\begin{eqnarray*}
\det (W_m) = a_{mm}W.
\end{eqnarray*}

(d) Conclude from part (a) and part (c) that

\begin{eqnarray*}
W' = (a_{11} + a_{22} + \cdots + a_{nn}W,
\end{eqnarray*}

and use this to show that if $$a<t_0<b$$ then

\begin{eqnarray*}
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.
\end{eqnarray*}

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

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 $$(a,b)$$.

(a) Show that $$Y(t_0)$$ is invertible.

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

\begin{eqnarray*}
{\bf y}' = A(t){\bf y}, \quad {\bf y}(t_0) = {\bf k}
\end{eqnarray*}

is

\begin{eqnarray*}
{\bf y} = Y(t)Y^{-1}(t_0){\bf k}.
\end{eqnarray*}

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

Let

\begin{eqnarray*}
A = \left[ \begin{array} \\ 2 & 4 \\ 4 & 2 \end{array} \right] \quad {\bf y}_1 = \left[ \begin{array} \\ e^{6t} \\ e^{6t} \end{array} \right], \quad {\bf y}_2 = \left[ \begin{array} \\ e^{-2t} \\ -e^{-2t} \end{array} \right], \quad {\bf k} = \left[ \begin{array} \\ -3 \\ 9 \end{array} \right].
\end{eqnarray*}

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

(b) Solve the initial value problem

\label{eq:4.3E.3}
{\bf y}' = A{\bf y}, \quad {\bf y}(0) = {\bf k}.

(c) Use the result of Exercise $$(4.3E.6)$$ part (b) to find a formula for the solution of \eqref{eq:4.3E.3} for an arbitrary initial vector $${\bf k}$$.

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

Repeat Exercise $$(4.3E7)$$ with

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

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

Repeat Exercise $$(4.3E.7)$$ with

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

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

Repeat Exercise $$(4.3E.7)$$ with

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

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

Let

\begin{eqnarray*}
A &=& \left[ \begin{array} \\ 3 & -1 & -1 \\ -2 & 3 & 2 \\ 4 & -1 & -2 \end{array} \right], \\ {\bf y}_1 &=& \left[ \begin{array} \\ e^{2t} \\ 0 \\ e^{2t} \end{array} \right], \quad {\bf y}_2 = \left[ \begin{array} \\ e^{3t} \\ -e^{3t} \\ e^{3t} \end{array} \right], \quad {\bf y}_3 = \left[ \begin{array} \\ e^{-t} \\ -3e^{-t} \\ 7e^{-t} \end{array} \right], \quad {\bf k} = \left[ \begin{array} \\ 2 \\ -7 \\ 20 \end{array} \right].
\end{eqnarray*}

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

(b) Solve the initial value problem

\label{eq:4.3E.4}
{\bf y}' = A{\bf y}, \quad {\bf y}(0) = {\bf k}.

(c) Use the result of Exercise $$(4.3E.6)$$ part (b) to find a formula for the solution of \eqref{eq:4.3E.4} for an arbitrary initial vector $${\bf k}$$.

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

Repeat Exercise $$(4.3E.11)$$ with

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

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### Exercise $$\PageIndex{13}$$

Repeat Exercise $$(4.3E.11)$$ with

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

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### Exercise $$\PageIndex{14}$$

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.

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### Exercise $$\PageIndex{15}$$

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.

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

(b) 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 $${\bf y}'=A(t){\bf y}$$.

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### Exercise $$\PageIndex{16}$$

Suppose the $$n\times n$$ matrix $$A=A(t)$$ is continuous on $$(a,b)$$ and $$t_0$$ is in $$(a,b)$$. For $$i=1$$, $$2$$, $$\dots$$, $$n$$, 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

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

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

(a) 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 $$(a,b)$$.

(b) Conclude from part (a) and Exercise $$(4.3E.15)$$ that $${\bf y}'=A(t){\bf y}$$ has infinitely many fundamental sets of solutions on $$(a,b)$$.

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### Exercise $$\PageIndex{17}$$

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 $$A$$.
Hint: See Exercise $$(4.2E.11)$$.

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### Exercise $$\PageIndex{18}$$

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

(a) Show that $$Z(t)Z(s)=Z(t+s)$$ for all $$s$$ and $$t$$.
Hint: For fixed $$s$$ let $$\Gamma_1(t)=Z(t)Z(s)$$ and $$\Gamma_2(t)=Z(t+s)$$. Show that $$\Gamma_1$$ and $$\Gamma_2$$ are both solutions of the matrix initial value problem $$\Gamma'=A\Gamma,\quad\Gamma(0)=Z(s)$$. Then conclude from Theorem $$(4.2.1)$$ that $$\Gamma_1=\Gamma_2$$.

(b) Show that $$(Z(t))^{-1}=Z(-t)$$.

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