For the system below, find the general solution, sketch the trajectories, being careful to include the eigenvector directions, and classify the type of fixed point:\[\dot{x} = -x + y, \quad \dot{y} = 4x - y.\]Consider a planar system in terms of eigenvalues and eigenvectors.

Convert given system into matrix. Find eigenvalues and corresponding eigenvectors. Find general solution Find fixed point from nature of eigenvalues Draw or sketch trajectories.

Answered: the system below, find the general…

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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For an electrical circuit containing an inductor, a resistor, and a capacitor, the charge Q(t) on the capacitor satisfies the ODE 8²³Q +2 dt² dQ dt +2 +2Q=0, (in suitably scaled variables). Write the equation as a system of first-order ODEs and find the general solution using the eigenvector method.
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1. For the system below, find the general solution, sketch the trajectories, being careful to include the eigenvector directions, and classify the type of fixed point: x = x, ÿ y = 2x - 5y.
(B. Janssen, KTH, 2014) Consider the linear system 0.550x+0.423y = 0.127 0.484x + 0.372y = 0.112 Suppose we are given two possible solutions, u = [_11] and v- -1.91. 1.01 0.9 a. Decide based on the residuals b - Au and b - Av which of the two possible solutions is the 'better' solution. b. Calculate the exact solution x. c. Compute the errors to the exact solution. That is, compute the infinity norms of u-x and v-x. Do the results change your answer to 7a?
f) Use your eigenvalues and eigenvectors to write the general solution, in matrix form, to your linear system from part (a). (Your answer should have two arbitrary constants). g) Apply your initial conditions from part (a) to solve for the arbitrary constants. Please answer a-h with clear steps thanks h) Take your solution out of matrix form and rewrite it in the form Q1= f(t), Q2 = g(t)
Find two linearly independent solutions of y″+2xy=0 of the form and enter the first few coefficients shown in the photo

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the system below, find the general solution, sketch the trajectories, bei lude the eigenvector directions, and classify the type of fixed point: x = -x + y, y = 4x - y.