10.3.1: Basic Theory of Homogeneous Linear Systems (Exercises)

Q10.3.1

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 $(a,b)$.

2. In Section 5.1 the Wronskian of two solutions $y_1$ and $y_2$ of the scalar second order equation

$$P_0(x)y''+P_1(x)y'+P_2(x)y=0 \tag{A}$$

was defined to be

$$W=\left|\begin{array}{cc} y_1&y_2 \\[4pt] y'_1&y'_2\end{array}\right|.\nonumber$$

1. 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.
2. Apply Equation 10.3.6 to the system derived in (a), and show that $$W(x)=W(x_0)\exp\left\{-\int^x_{x_0}{P_1(s)\over P_0(s)}\, ds\right\},\nonumber$$ which is the form of Abel’s formula given in Theorem 9.1.3.

3. In Section 9.1 the Wronskian of $n$ solutions $y_1$, $y_2$, …, $y_n$ of the $n-$th order equation

$$P_0(x)y^{(n)}+P_1(x)y^{(n-1)}+\cdots+P_n(x)y=0 \tag{A}$$

was defined to be

$$W=\left|\begin{array}{cccc} y_1&y_2&\cdots&y_n \\[4pt] y'_1&y'_2&\cdots&y_n'\\[4pt] \vdots&\vdots&\ddots&\vdots\\[4pt] y_1^{(n-1)}&y_2^{(n-1)}&\cdots&y_n^{(n-1)} \end{array}\right|.\nonumber$$

1. 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.
2. Apply Equation 10.3.6 to the system derived in (a), and show that $$W(x)=W(x_0)\exp\left\{-\int^x_{x_0}{P_1(s)\over P_0(s)}\, ds\right\},\nonumber$$ which is the form of Abel’s formula given in Theorem 9.1.3.

4. Suppose

$${\bf y}_1=\left[\begin{array}{c}{y_{11}}\\[4pt]{y_{21}}\end{array} \right]\quad\text{and}\quad {\bf y}_2=\left[\begin{array}{c}{y_{12}}\\[4pt]{y_{22}}\end{array} \right]\nonumber$$

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

$$Y=\left[\begin{array}{cc}{y_{11}}&{y_{12}}\\[4pt]{y_{21}}&{y_{22}}\end{array} \right]\quad\text{and}\quad W=\left|\begin{array}{cc}{y_{11}}&{y_{12}}\\[4pt]{y_{21}}&{y_{22}}\end{array} \right|\nonumber$$

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

1. Deduce from the definition of determinant that $$W'=\left|\begin{array}{cc} {y'_{11}}&{y'_{12}}\\[4pt] {y_{21}}& {y_{22}}\end{array}\right| +\left|\begin{array}{cc} {y_{11}}&{y_{12}}\\[4pt] {y'_{21}}&{y'_{22}}\end{array}\right|.\nonumber$$
2. Use the equation $Y'=A(t)Y$ and the definition of matrix multiplication to show that $$[y'_{11}\quad y'_{12}]=a_{11} [y_{11}\quad y_{12}]+a_{12} [y_{21} \quad y_{22}]\nonumber$$ and $$[y'_{21}\quad y'_{22}]=a_{21} [y_{11}\quad y_{12}]+a_{22} [y_{21}\quad y_{22}].\nonumber$$
3. Use properties of determinants to deduce from (a) and (a) that $$\left|\begin{array}{cc} {y'_{11}}&{y'_{12}}\\[4pt] {y_{21}}& {y_{22}}\end{array}\right|=a_{11}W\quad \text{and} \quad \left|\begin{array}{cc} {y_{11}}&{y_{12}}\\[4pt] {y'_{21}}&{y'_{22}}\end{array}\right|=a_{22}W.\nonumber$$
4. Conclude from (c) that $$W'=(a_{11}+a_{22})W,\nonumber$$ and use this to show that if $a<t_0<b$ then $$W(t)=W(t_0)\exp\left(\int^t_{t_0} \left[a_{11}(s)+a_{22} (s) \right]\, ds\right)\quad a<t<b.\nonumber$$

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

$$Y= \left[\begin{array}{cccc} y_{11}&y_{12}&\cdots&y_{1n} \\[4pt] y_{21}&y_{22}&\cdots&y_{2n} \\[4pt] \vdots&\vdots&\ddots&\vdots \\[4pt] y_{n1}&y_{n2}&\cdots&y_{nn} \end{array}\right],\nonumber$$

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

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

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

1. Deduce from the definition of determinant that $$W'=W_1+W_2+\cdots+W_n,\nonumber$$ 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 $i=m$.
2. Use the equation $Y'=A Y$ and the definition of matrix multiplication to show that $$r'_m=a_{m1}r_1+a_{m2} r_2+\cdots+a_{mn}r_n.\nonumber$$
3. Use properties of determinants to deduce from (b) that $$\det (W_m)=a_{mm}W.\nonumber$$
4. Conclude from (a) and (c) that $$W'=(a_{11}+a_{22}+\cdots+a_{nn})W,\nonumber$$ and use this to show that if $a<t_0<b$ then $$W(t)=W(t_0)\exp\left( \int^t_{t_0}\big[a_{11}(s)+a_{22}(s)+\cdots+a_{nn}(s)]\, ds\right), \quad a < t < b.\nonumber$$

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 $(a,b)$.

1. Show that $Y(t_0)$ is invertible.
2. Show that if ${\bf k}$ is an arbitrary $n$-vector then the solution of the initial value problem $${\bf y}'=A(t){\bf y},\quad {\bf y}(t_0)={\bf k}\nonumber$$ is $${\bf y}=Y(t)Y^{-1}(t_0){\bf k}.\nonumber$$

7. Let

$$A=\left[\begin{array}{cc}{2}&{4}\\[4pt]{4}&{2}\end{array} \right], \quad {\bf y}_1=\left[\begin{array}{c} e^{6t} \\[4pt] e^{6t} \end{array}\right], \quad {\bf y}_2=\left[\begin{array}{r} e^{-2t} \\[4pt] -e^{-2t}\end{array}\right], \quad {\bf k}=\left[\begin{array}{r}-3 \\[4pt] 9\end{array}\right].\nonumber$$

1. Verify that $\{{\bf y}_1,{\bf y}_2\}$ is a fundamental set of solutions for ${\bf y}'=A{\bf y}$.
2. Solve the initial value problem $${\bf y}'=A{\bf y},\quad {\bf y}(0)={\bf k}. \tag{A}$$
3. Use the result of Exercise 10.3.6 (b) to find a formula for the solution of (A) for an arbitrary initial vector ${\bf k}$.

8. Repeat Exercise 10.3.7 with

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

9. Repeat Exercise 10.3.7 with

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

10. Repeat Exercise 10.3.7 with

$$A=\left[\begin{array}{cc}{2}&{1}\\[4pt]{1}&{2}\end{array} \right], \quad {\bf y}_1=\left[\begin{array}{r} e^{3t} \\[4pt] e^{3t} \end{array}\right], \quad {\bf y}_2=\left[\begin{array}{r}e^t \\[4pt] -e^t\end{array}\right], \quad {\bf k}=\left[\begin{array}{r} 2 \\[4pt] 8 \end{array}\right].\nonumber$$

11. Let

$$\begin{aligned} A&= \left[\begin{array}{ccc}{3}&{-1}&{-1}\\[4pt]{-2}&{3}&{2}\\[4pt]{4}&{-1}&{-2}\end{array} \right] , \\[4pt] {\bf y}_1&=\left[\begin{array}{c} e^{2t} \\[4pt] 0 \\[4pt] e^{2t}\end{array} \right], \quad {\bf y}_2=\left[\begin{array}{c} e^{3t} \\[4pt]-e^{3t} \\[4pt] e^{3t}\end{array}\right], \quad {\bf y}_3=\left[\begin{array}{c} e^{-t} \\[4pt]-3e^{-t} \\[4pt] 7e^{-t} \end{array}\right], \quad {\bf k}=\left[\begin{array}{r} 2 \\[4pt]-7 \\[4pt] 20\end{array}\right].\end{aligned}\nonumber$$

1. Verify that $\{{\bf y}_1,{\bf y}_2,{\bf y}_3\}$ is a fundamental set of solutions for ${\bf y}'=A{\bf y}$.
2. Solve the initial value problem $${\bf y}'=A{\bf y}, \quad {\bf y}(0)={\bf k}. \tag{A}$$
3. Use the result of Exercise 10.3.6 (b) to find a formula for the solution of (A) for an arbitrary initial vector ${\bf k}$.

12. Repeat Exercise 10.3.11 with

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

13. Repeat Exercise 10.3.11 with

$$\begin{aligned} A&=\left[\begin{array}{ccc}{-1}&{2}&{3}\\[4pt]{0}&{1}&{6}\\[4pt]{0}&{0}&{-2}\end{array} \right], \\[4pt] {\bf y}_1&=\left[\begin{array}{c} e^t \\[4pt] e^t \\[4pt] 0\end{array}\right], \quad {\bf y}_2=\left[\begin{array}{c} e^{-t} \\[4pt] 0 \\[4pt] 0\end{array}\right], \quad {\bf y}_3=\left[\begin{array}{c} e^{-2t} \\[4pt]-2e^{-2t} \\[4pt] e^{-2t}\end{array}\right], \quad {\bf k}=\left[\begin{array}{r} 5 \\[4pt] 5 \\[4pt]-1 \end{array}\right].\end{aligned}\nonumber$$

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.

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.

1. Show that the matrix $Z=YC$ satisfies the differential equation $Z'=AZ$.
2. 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 ${\bf y}'=A(t){\bf y}$.

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

$${\bf e}_1=\left[\begin{array}{c} 1\\[4pt]0\\[4pt] \vdots\\[4pt]0\end{array}\right],\quad {\bf e}_2=\left[\begin{array}{c} 0\\[4pt]1\\[4pt] \vdots\\[4pt]0\end{array}\right],\quad\cdots\quad {\bf e}_n=\left[\begin{array}{c} 0\\[4pt]0\\[4pt] \vdots\\[4pt]1\end{array}\right];\nonumber$$

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

1. 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 $(a,b)$.
2. Conclude from (a) and Exercise 10.3.15 that ${\bf y}'= A(t){\bf y}$ has infinitely many fundamental sets of solutions on $(a,b)$.

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 10.3.11.

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

1. 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 $\Gamma '=A\Gamma , \:\Gamma (0)=Z(s)$. Then conclude from Theorem 10.2.1 that $\Gamma _{1}=\Gamma _{2}$.
2. Show that $(Z(t))^{-1}=Z(-t)$.
3. The matrix $Z$ defined above is sometimes denoted by $e^{tA}$. Discuss the motivation for this notation.