**Consider the System**Given the differential equation system:\[\vec{x}' = \begin{pmatrix} 2 & -5 \\ 1 & -2 \end{pmatrix} \vec{x}.\]First, verify that:\[\vec{x}^{(1)} = \begin{pmatrix} 5 \cos(t) \\ 2 \cos(t) + \sin(t) \end{pmatrix}\]and\[\vec{x}^{(2)} = \begin{pmatrix} 5 \sin(t) \\ 2 \sin(t) - \cos(t) \end{pmatrix}\]are solutions of the system. Then:(a) Show that \(\vec{x} = c_1 \vec{x}^{(1)} + c_2 \vec{x}^{(2)}\) is also a solution of the system for arbitrary \(c_1\) and \(c_2\).(b) Show that \(\vec{x}^{(1)}\) and \(\vec{x}^{(2)}\) form a fundamental set of solutions of the system.(c) Find a solution of the system that satisfies the initial condition:\[\vec{x}(0) = \begin{pmatrix} 1 \\ 2 \end{pmatrix}.\](d) Find \(W(t) = W[\vec{x}^{(1)}, \vec{x}^{(2)}](t)\).(e) Show that \(W = W(t)\) satisfies the Abel's equation:\[W' = (p_{11}(t) + p_{22}(t))W.\]

Consider the system 7-()7 2 -5 1 First verify that 5 cos(t) 70 = 2 cos(t) + sin(t) 79 = 5 sin(t) (2sin(t) – cos(t) ) and are solutions of the system. Then (a) show that = c7(1) + c2 T (2) is also a solution of the system for arbitrary c and C2; (b) show that 7(1) and (2) form a fundamental set of solutions of the system; (c) find a solution of the system that satisfies the initial condition 국 (0) = (d) find W(t) = W[70, 7](t); (e) show that W = W(t) satisfies the Abel's equation %3D W' = (P11(t) + P22 (t))W.

Consider the system 7-()7 2 -5 1 First verify that 5 cos(t) 70 = 2 cos(t) + sin(t) 79 = 5 sin(t) (2sin(t) – cos(t) ) and are solutions of the system. Then (a) show that = c7(1) + c2 T (2) is also a solution of the system for arbitrary c and C2; (b) show that 7(1) and (2) form a fundamental set of solutions of the system; (c) find a solution of the system that satisfies the initial condition 국 (0) = (d) find W(t) = W[70, 7](t); (e) show that W = W(t) satisfies the Abel's equation %3D W' = (P11(t) + P22 (t))W.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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