Question 1:Find the general solution of x' = Ax with A =and eigenvectors are given belowA₁ = 1, v(¹)=-1(?);11₂ = 2, v (²)=(²)31-2 -42; X3 = 3, v (³)=211if the eigenvalues0(G)

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Answered: Question 1: and eigenvectors are given…

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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4 Find A" where A = You are given that the eigenvalues of A are 3 and 4 with corresponding eigenvectors -2 3 and 1 [4" 2(4"-3")] 0. 3" 4h 0. (-2)" 3" 0. 2(3-4") 3" || 3n 0. 2(4-3") 4" None of these.
1-3 2) 2) Verify that F = 0 -4 3 has eigenvalue/eigenvector pairs 2, 0-6 5) and compute D = Q¯'FQ. (1 2 1 20.₁.0-¹.0 () Write Q=1 0 2 20 2
3 *. Sketch a graph of the eigenvectors and plot Solve the system of ODES given by i' = | , some sample trajectories. Is the origin an attractor, a repeller or a saddle point? -2.
1(1)
The eigenvalues of the initial value problem y' +Ây=0, y(0)=0, y'(7)=0 are a) , = vn, for n=1,2,3,... b) 1, =n², for n=1,2,3,... c) 2, = 2n, for n=1,2,3,... d) , =, for n=1,2,3,... 2'
The eigenfunctions for y" + 2y' + Ay = y(0) = 1, y(1) = 0 are {e" sin(nπa)O {e2r sin(프플프) 1O n=1 {e-"sin(뿌)EO od In=1 4 II {e-2" sin(nπa)이 {e" sin ()O 2 n=1 .2r -:.. ( po
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Q3. (a) If Ax = 2 x, determine the eigenvalues and the corresponding eigenvectors for A= G ) 2 3. Hence, write down the associated modal matrix P and diagonal matrix D, and use these values to solve the following system differential equations: *1 = X1 + 4 x2 X2 = 2 x1 + 3 x2 Given that when t = 0, x1 = 0 and x2 = 2. (b) Use the 4th order Runge Kutta method to solve the differential equation: dy e* y dx for values of x 0 (0.2) 0.4 given that y 1 when x 0. %3D Give your answers correct to 5 decimal places. Obtain the analytical solution of the differential equation and compare the analytical solution when x = 0.4 with the values obtained using Runge Kutta,
Solve the system of ODES given by x' x. Sketch a graph of the eigenvectors and plot 13 some sample trajectories. Is the origin an attractor, a repeller or a saddle point? Give your final solution in exact form with e.

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Question 1: and eigenvectors are given below Find the general solution of x' = Ax with A = -(;'); x-²-(1) A₂ = 2,v(2) A₁ = 1, v(¹1) - -2 0 3 1 2 4 2 1 -1 ; A3 = 3, v (³); if the eigenvalues -(6)