3. Obtain the eigenvalues and eigenfunctions of the following Sturm-Liouville systems:(a) y" + y/ + (1 + A) y = 0,y (0) = 0, y (1) = 0.%3D(c) y" - 3y' +3 (1+ A) y = 0,%3Dy' (0) = 0, y' (7) = 0.

Given differential equation is y''+y'+(1+λ)y=0, y(0)=0=y(1) Case 1: For λ=0 we have…

Answered: 3. Obtain the eigenvalues and…

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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7. Suppose 2 x 2 matrix A has an eigenvalue i with eigenvector =[¹-1]. (a) Find the general solution of = A. (b) Draw the phase portrait and classify the equilibrium solution with its stability. You must show the work behind your drawing.
Consider the Sturm-Liouville problem: y" + Ay /(-L) 0, -L
The eigenvalues of the system (x +1]y" + 2xy' - 4y = 0 are A. None of them B. x = -i and x = i C. Å1 = -3 and Az = 4 %3D D. A1 = -4 and Å = 2 O E. x = -1 and x = 1 %3D
the given figure is a phase portrait for a system x' = Ax, (a) The eigenvalues of A could be (circle one) i. A1 = 1, A2 = 2 ii. A1 = -1, A2 = -2 iii. A1 = -1, A2 = 2 iv. A1,2 = 1+i v. A1,2 = -1±i (b) If possible, give one eigenvector of A (if it is not possible, write "N/A"): %3D Answer:
You are solving a 2x2 system of the form : y' = PỶ + where P has constant entries with real different eigenvalues. Decide which of the following statements is correct: I can find a particular solution using the undetermined coefficients method with a try function of the form Y(t) = sin(t)å + cos(t)b I can find a particular solution using the undetermined coefficients method with a try function of the form Y(t) = (cos(t) + sin(t))ā I can only find the general solution for this problems using the Variation of Parameters method. Undetermined Coefficients won't work in this case. I can find the general solution using the undetermined coefficients method. To find the particular solution I should use a with a try function of the form Y(t) = sin(t)ä
6. The nonlinear system y₁ = -y₁ - y2, y₂ = y₁ + y₂, has steady states at (0,0), (1, -1), and (-1,1). Classify each steady state as a source (all eigenvalues have positive real part), sink (all eigenvalues have negative real part), saddle (one eigenvalue with positive real part and one with negative real part), or center (all eigenvalues purely imaginary).
12- The system x = x + 4y, y' = y + x has two eigenvectors....
Let A be a 2 x 2 matrix with eigenvalues X₁ = -3 and >₂ = -1 and [1] an H Let x(t) be the position of a particle at time t. Suppose we solved the initial corresponding eigenvectors V₁ = value problem x = Ax, x(0) = [3] -ce + c₂e-t x(t) = = [2(t)] = [ qe * +ge + -3t Search and C2= -3t and V2 = to obtain then C1=
Suppose that a 2 x 2 matrix A with real entries has an eigenvalue X = 1 + 3i with corresponding 4 i [12] eigenvector 1 + 2i Which of the following is NOT a solution of the linear system y' = Ay? y(t) = et y(t) = et y(t) = et y(t) = et 4 cos(3t) sin(3t) [cos(3t) + 2 sin(3t). 4 cos(3t) + sin(3t) cos(3t) 2 sin (3t) - cos(3t) + 4 sin(3t) 2 cos(3t) + sin(3t) cos (3t) - 4 sin (3t) -2 cos(3t) - sin(3t)
I already know the eigenvalues are -2+6i, and 2-6i, and the eigenvector is [6i,1] , and [-6i,1], im just struggling finding the overall solution
1b
can someone explain this for me, please ?

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3. Obtain the eigenvalues and eigenfunctions of the following Sturm- Liouville systems: (a) y"+y' + (1+A) y = 0, y (0) = 0, y (1) = 0.