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
\[W=\left|\begin{array}{cc} y_1&y_2 \\[4pt] y'_1&y'_2\end{array}\right|.\nonumber \]
- 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 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
\[W=\left|\begin{array}{cccc} y_1&y_2&\cdots&y_n \\[4pt] y'_1&y'_2&\cdots&y_n'\\[4pt] \vdots&\vdots&\ddots&\vdots\\[4pt] y_1^{(n-1)}&y_2^{(n-1)}&\cdots&y_n^{(n-1)} \end{array}\right|.\nonumber \]
- 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 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 \(\{{\bf y}_1,{\bf y}_2\}\).
- Deduce from the definition of determinant that \[W'=\left|\begin{array}{cc} {y'_{11}}&{y'_{12}}\\[4pt] {y_{21}}& {y_{22}}\end{array}\right| +\left|\begin{array}{cc} {y_{11}}&{y_{12}}\\[4pt] {y'_{21}}&{y'_{22}}\end{array}\right|.\nonumber \]
- 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 \[[y'_{21}\quad y'_{22}]=a_{21} [y_{11}\quad y_{12}]+a_{22} [y_{21}\quad y_{22}].\nonumber \]
- Use properties of determinants to deduce from (a) and (a) that \[\left|\begin{array}{cc} {y'_{11}}&{y'_{12}}\\[4pt] {y_{21}}& {y_{22}}\end{array}\right|=a_{11}W\quad \text{and} \quad \left|\begin{array}{cc} {y_{11}}&{y_{12}}\\[4pt] {y'_{21}}&{y'_{22}}\end{array}\right|=a_{22}W.\nonumber \]
- 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 \(Y\).
- 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 \(i=m\).
- Use the equation \(Y'=A Y\) and the definition of matrix multiplication to show that \[r'_m=a_{m1}r_1+a_{m2} r_2+\cdots+a_{mn}r_n.\nonumber \]
- Use properties of determinants to deduce from (b) that \[\det (W_m)=a_{mm}W.\nonumber \]
- 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 \((a,b)\).
- 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 \[{\bf y}'=A(t){\bf y},\quad {\bf y}(t_0)={\bf k}\nonumber \] is \[{\bf y}=Y(t)Y^{-1}(t_0){\bf k}.\nonumber \]
7. Let
\[A=\left[\begin{array}{cc}{2}&{4}\\[4pt]{4}&{2}\end{array} \right], \quad {\bf y}_1=\left[\begin{array}{c} e^{6t} \\[4pt] e^{6t} \end{array}\right], \quad {\bf y}_2=\left[\begin{array}{r} e^{-2t} \\[4pt] -e^{-2t}\end{array}\right], \quad {\bf k}=\left[\begin{array}{r}-3 \\[4pt] 9\end{array}\right].\nonumber \]
- 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 \[{\bf y}'=A{\bf y},\quad {\bf y}(0)={\bf k}. \tag{A} \]
- 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
\[\begin{aligned} A&= \left[\begin{array}{ccc}{3}&{-1}&{-1}\\[4pt]{-2}&{3}&{2}\\[4pt]{4}&{-1}&{-2}\end{array} \right] , \\[4pt] {\bf y}_1&=\left[\begin{array}{c} e^{2t} \\[4pt] 0 \\[4pt] e^{2t}\end{array} \right], \quad {\bf y}_2=\left[\begin{array}{c} e^{3t} \\[4pt]-e^{3t} \\[4pt] e^{3t}\end{array}\right], \quad {\bf y}_3=\left[\begin{array}{c} e^{-t} \\[4pt]-3e^{-t} \\[4pt] 7e^{-t} \end{array}\right], \quad {\bf k}=\left[\begin{array}{r} 2 \\[4pt]-7 \\[4pt] 20\end{array}\right].\end{aligned}\nonumber \]
- 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 \[{\bf y}'=A{\bf y}, \quad {\bf y}(0)={\bf k}. \tag{A} \]
- 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.
- 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 \({\bf y}'=A(t){\bf y}\).
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 \(j\ne i\).
- 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 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 \(Z(0)=I\).
- 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 \(\Gamma _{1}=\Gamma _{2}\).
- Show that \((Z(t))^{-1}=Z(-t)\).
- The matrix \(Z\) defined above is sometimes denoted by \(e^{tA}\). Discuss the motivation for this notation.
