Solution Found!
In this problem we outline a proof of Theorem 7.4.3 in the
Chapter 7, Problem 2(choose chapter or problem)
In this problem we outline a proof of Theorem 7.4.3 in the case n = 2. Let \(\mathbf{x}^{(1)}\) and \(\mathbf{x}^{(2)}\) be solutions of Eq. (3) for \(\alpha<t<\beta\), and let W be the Wronskian of \(\mathbf{x}^{(1)}\) and \(\mathbf{x}^{(2)}\).
(a) Show that
\(\frac{d W}{d t}=\left|\begin{array}{cc}
\frac{d x_{1}^{(1)}}{d t} & \frac{d x_{1}^{(2)}}{d t} \\
x_{2}^{(1)} & x_{2}^{(2)}
\end{array}\right|+\left|\begin{array}{cc}
x_{1}^{(1)} & x_{1}^{(2)} \\
\frac{d x_{2}^{(1)}}{d t} & \frac{d x_{2}^{(2)}}{d t}
\end{array}\right|\)
(b) Using Eq. (3), show that
\(\frac{d W}{d t}=\left(p_{11}+p_{22}\right) W\).
(c) Find W(t) by solving the differential equation obtained in part (b). Use this expression to obtain the conclusion stated in Theorem 7.4.3.
(d) Prove Theorem 7.4.3 for an arbitrary value of n by generalizing the procedure of parts (a), (b), and (c).
Questions & Answers
QUESTION:
In this problem we outline a proof of Theorem 7.4.3 in the case n = 2. Let \(\mathbf{x}^{(1)}\) and \(\mathbf{x}^{(2)}\) be solutions of Eq. (3) for \(\alpha<t<\beta\), and let W be the Wronskian of \(\mathbf{x}^{(1)}\) and \(\mathbf{x}^{(2)}\).
(a) Show that
\(\frac{d W}{d t}=\left|\begin{array}{cc}
\frac{d x_{1}^{(1)}}{d t} & \frac{d x_{1}^{(2)}}{d t} \\
x_{2}^{(1)} & x_{2}^{(2)}
\end{array}\right|+\left|\begin{array}{cc}
x_{1}^{(1)} & x_{1}^{(2)} \\
\frac{d x_{2}^{(1)}}{d t} & \frac{d x_{2}^{(2)}}{d t}
\end{array}\right|\)
(b) Using Eq. (3), show that
\(\frac{d W}{d t}=\left(p_{11}+p_{22}\right) W\).
(c) Find W(t) by solving the differential equation obtained in part (b). Use this expression to obtain the conclusion stated in Theorem 7.4.3.
(d) Prove Theorem 7.4.3 for an arbitrary value of n by generalizing the procedure of parts (a), (b), and (c).
ANSWER:Step 1 of 6
a) The homogeneous equation is \(\mathbf{x}^{\prime}=\mathbf{P}(t) \mathbf{x}\).
Assume that the solutions of the system \(\mathbf{x}^{\prime}=\mathbf{P}(t) \mathbf{x}\) are \(\mathbf{x}^{(1)}\) and \(\mathbf{x}^{(2)}\) for \(\alpha<t<\beta\).
The objective is to determine \(\frac{d W}{d t}\).
The Wronskian of the solutions \(\mathbf{x}^{(1)}\) and \(\mathbf{x}^{(2)}\) is \(W=\operatorname{det}\left[\begin{array}{ll}
\mathbf{x}^{(1)} & \mathbf{x}^{(2)}
\end{array}\right]\)
\(\mathbf{x}^{(1)}=\left[\begin{array}{l}
x_{1}^{(1)} \\
x_{2}^{(1)}
\end{array}\right] \text { and } \mathbf{x}^{(2)}=\left[\begin{array}{c}
x_{1}^{(2)} \\
x_{2}^{(2)}
\end{array}\right]\)
The Wronskian of \(\mathbf{x}^{(1)}\) and \(\mathbf{x}^{(2)}\) is
\(\begin{aligned}
W\left(\mathbf{x}^{(1)}(t), \mathbf{x}^{(2)}(t)\right) & =\left|\begin{array}{ll}
x_{1}^{(1)} & x_{1}^{(2)} \\
x_{2}^{(1)} & x_{2}^{(2)}
\end{array}\right| \\
& =W
\end{aligned}\)