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

• • Contributed by William F. Trench
• Andrew G. Cowles Distinguished Professor Emeritus (Mathamatics) at Trinity University
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## 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

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

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

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

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

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

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

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

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

1. Deduce from the definition of determinant that $W'=\left|\begin{array}{cc} {y'_{11}}&{y'_{12}}\\ {y_{21}}& {y_{22}}\end{array}\right| +\left|\begin{array}{cc} {y_{11}}&{y_{12}}\\ {y'_{21}}&{y'_{22}}\end{array}\right|.\nonumber$
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 $\left|\begin{array}{cc} {y'_{11}}&{y'_{12}}\\ {y_{21}}& {y_{22}}\end{array}\right|=a_{11}W\quad \text{and} \quad \left|\begin{array}{cc} {y_{11}}&{y_{12}}\\ {y'_{21}}&{y'_{22}}\end{array}\right|=a_{22}W.\nonumber$
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

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

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

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

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

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

9. Repeat Exercise 10.3.7 with

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

10. Repeat Exercise 10.3.7 with

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

11. Let

\begin{aligned} A&= \left[\begin{array}{ccc}{3}&{-1}&{-1}\\{-2}&{3}&{2}\\{4}&{-1}&{-2}\end{array} \right] , \\ {\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}\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}\\{2}&{0}&{2}\\{2}&{2}&{0}\end{array} \right], \\ {\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}\nonumber

13. Repeat Exercise 10.3.11 with

\begin{aligned} A&=\left[\begin{array}{ccc}{-1}&{2}&{3}\\{0}&{1}&{6}\\{0}&{0}&{-2}\end{array} \right], \\ {\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}\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

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

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.