91Ó°ÊÓ

\(\mathbf{X}^{\prime}=\left(\begin{array}{ll}2 & -5 \\ 1 & -2\end{array}\right) \mathbf{X}+\mathbf{F}(t)\) for (a) \(\mathbf{F}(t)=0\) (b) \(\mathbf{F}(t)=\left(\begin{array}{c}-2 \cos t+4 \sin t \\ 2 \sin t\end{array}\right)\)

\(x^{\prime}=2 y, y^{\prime}=2 \sin 2 x ; R=[-\pi, \pi] \times[-\pi, \pi]\)

(Principle of Superposition) (a) Show that any linear combination of solutions of the homogeneous system \(\mathbf{X}^{\prime}(t)=\mathbf{A}(t) \mathbf{X}(t)\) is also a solution of the homogeneous system. (b) Is the Principle of Superposition ever valid for nonhomogeneous systems of equations? Explain.

\(\mathbf{X}^{\prime}=\left(\begin{array}{ll}3 & -1 \\ 4 & -1\end{array}\right) \mathbf{X}+\left(\begin{array}{c}\cos t \\ \sin t\end{array}\right), \mathbf{X}(0)=\left(\begin{array}{l}0 \\\ 0\end{array}\right)\)

(a) Find a general solution of \(\mathbf{X}^{\prime}=\mathbf{A X}\) if \(\mathbf{X}(t)=\left(\begin{array}{l}x_{1} \\ x_{2} \\ x_{3} \\\ x_{4}\end{array}\right)\) and (i) \(\mathbf{A}=\left(\begin{array}{cccc}\lambda & 0 & 0 & 0 \\ 0 & \lambda & 0 & 0 \\ 0 & 0 & \lambda & 0 \\ 0 & 0 & 0 & \lambda\end{array}\right)\) (ii) \(\mathbf{A}=\left(\begin{array}{cccc}\lambda & 1 & 0 & 0 \\ 0 & \lambda & 0 & 0 \\ 0 & 0 & \lambda & 0 \\ 0 & 0 & 0 & \lambda\end{array}\right)\), (iii) \(\mathbf{A}=\left(\begin{array}{llll}\lambda & 1 & 0 & 0 \\ 0 & \lambda & 1 & 0 \\ 0 & 0 & \lambda & 0 \\ 0 & 0 & 0 & \lambda\end{array}\right)\), and (iv) \(\mathbf{A}=\left(\begin{array}{cccc}\lambda & 1 & 0 & 0 \\ 0 & \lambda & 1 & 0 \\ 0 & 0 & \lambda & 1 \\ 0 & 0 & 0 & \lambda\end{array}\right)\). (b) For each system in (a), find the solution that satisfies the initial condition \(\mathbf{X}(0)=\left(\begin{array}{c}-1 \\ 0 \\ 1 \\\ 2\end{array}\right)\) if \(\lambda=-1 / 2\) and then graph \(x_{1}, x_{2}, x_{3}\), and \(x_{4}\) for \(0 \leq t \leq 10\). How are the solutions similar? How are they different? (c) Indicate how to generalize the results obtained in (a). How would you find a general solution of \(\mathbf{X}^{\prime}=\mathbf{A} \mathbf{X}\) for the \(5 \times 5\) matrix \(\mathbf{A}=\) \(\left(\begin{array}{lllll}\lambda & 1 & 0 & 0 & 0 \\ 0 & \lambda & 1 & 0 & 0 \\\ 0 & 0 & \lambda & 1 & 0 \\ 0 & 0 & 0 & \lambda & 1 \\ 0 & 0 & 0 & 0 & \lambda\end{array}\right)\) ? How would you find a general solution of \(\mathbf{X}^{\prime}=\mathbf{A X}\) for the \(n \times n\) $$ \text { matrix } \mathbf{A}=\left(\begin{array}{cccccc} \lambda & 1 & 0 & \cdots & 0 & 0 \\ 0 & \lambda & 1 & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \ddots & \vdots & \vdots \\ \vdots & \vdots & \vdots & \ddots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & \lambda & 1 \\ 0 & 0 & 0 & \cdots & 0 & \lambda \end{array}\right) ? $$ Let \(\mathbf{X}=\mathbf{X}(t)=\left(\begin{array}{c}x_{1}(t) \\ x_{2}(t) \\\ \vdots \\ x_{n}(t)\end{array}\right), \mathbf{A}=\mathbf{A}(t)=\) \(\left(\begin{array}{cccc}a_{11}(t) & a_{12}(t) & \cdots & a_{1 n}(t) \\\ a_{21}(t) & a_{22}(t) & \cdots & a_{2 n}(t) \\ \vdots & \vdots & \ddots & \vdots \\ a_{n 1}(t) & a_{n 2}(t) & \cdots & a_{n n}(t)\end{array}\right), \mathbf{F}(t)=\left(\begin{array}{c}f_{1}(t) \\ f_{2}(t) \\ \vdots \\\ f_{n}(t)\end{array}\right)\), and \(\boldsymbol{\Phi}(t)\) be a fundamental matrix of the corresponding homogeneous system \(\mathbf{X}^{\prime}=\mathbf{A X}\). Then a general solution to the corresponding homogeneous system \(\mathbf{X}^{\prime}=\mathbf{A X}\) is \(\mathbf{X}_{h}=\boldsymbol{\Phi}(t) \mathbf{C}\) where \(\mathbf{C}=\left(\begin{array}{c}c_{1} \\ c_{2} \\ \vdots \\\ c_{n}\end{array}\right)\) is an \(n \times 1\) constant matrix. To find a general solution to the linear nonhomogeneous system \(\mathbf{X}^{\prime}=\mathbf{A} \mathbf{X}+\mathbf{F}(t)\), we proceed in the same way we did with linear nonhomogeneous equations in Chapter 4 . If \(\mathbf{X}_{p}(t)\) is a particular solution of the nonhomogeneous system, then all other solutions \(\mathbf{X}\) of the system can be written in the form \(\mathbf{X}(t)=\mathbf{X}_{h}(t)+\mathbf{X}_{p}(t)=\) \(\boldsymbol{\Phi}(t) \mathbf{C}+\mathbf{X}_{p}(t)\) (see Exercise 36).