Consider the problem:y″ + 2y′ + (1 − \)y = 0, y'(0) = y' (π) = 01. Find the eigenvalues and eigenfunctions.2. Verify that the eigenfunctions of the problem are orthogonal on [0,7]to the weight function w(x) = −1.3. Find an orthonormnal set of eigenfunction on [0, 7].

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Advanced Engineering Mathematics

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6. Consider the eigenvalue problem y" + ày = 0; y'(0) = 0, y(1) + y'(1) = 0. All the eigenvalues are nonnegative, so write à = a² where a 2 0. (a) Show that A = 0 is not an eigen- value. (b) Show that y = Acos ax + B sin ax satis- fies the endpoint conditions if and only if B = 0 and a is a positive root of the equation tan z = 1/z. These roots {an}° are the abscissas of the points of intersection of the curves y = tan z and y = 1/z, as indicated in Fig. 3.8.13. Thus the eigenvalues and eigenfunctions of this problem are the numbers {a;}° and the functions {cos an x}9°, re- spectively. y = 2n Зл I ly = tan z
Consider the Sturm-Liouville problem: y" + Ay /(-L) 0, -L
Solve the eigenvalue problems: Ly ₁²y = xy dr² for the two following sets of boundary conditions i. y(0)) = 0 and y(3) = 0 Ak = Yk (x) ii. y(0)) = 0 and y' (3) = 0 Ak = Yk (x) **NOTE: you must leave your answers for the eigenfunctions written in terms of X₁, rather than writing them out explicitly in terms of k, otherwise they won't be recognised as correct***
Find the smallest eigen value of (1+x)y" + y' + λy = 0, y(0)=0, y(1)=0
Exercise 4. (a) Suppose T E L(C²) is defined by T(x, y) = (-4y, x). Find the eigenvalues and the singular values of T. (b) Suppose T E L(C*) is defined by T(z1, 2, z3, 24) = (2, 2z3, 324, 0). Find the eigenvalues and the singular values of T.
What are the eigenvalues and eigenfunctions: x′′ + λx = 0, x(1) = x(3), x′(1) = x′(3) - Definey by x(z) = y((z−2)π) for 1 ≤ z ≤ 3. Show that y(−π) = y(π) and y′(−π) = y′(π) - Substitute into the equation to get y′′((z−2)π) + (λ/π2) y((z−2)π), for 1 ≤ z ≤ 3 - Use the change of variable t = (z − 2)π to show that the above equation has a non-zero solution if and only if either λ = k2π2 for some integer k ≥ 1 or λ = 0 and the solutions (eigenfunctions) are given by cos(kt), sin(kt) and 1 for −π ≤ t ≤ π. - Plug back t = (z − 2)π to find the eigenfunctions and eigenvalues of original equation
6. (a) Let T : L²[0, 2π] → L²[0, 2ñ] be given by 2π Tf(x) = s cos(xt) f(t) dt. Show that (i) T is self-adjoint. (ii) cos x and sin x are eigenvectors of T.
3. Consider the eigenvalue/boundary value problem for y(t): − 3y" + xy = 0, y′(0) = 0, y' (√3)= 0 (a) Is A = 0 an eigenvalue? If it is, calculate the corresponding eigenfunctions. (b) Determine all negative eigenvalues, A< 0, and calculate the corresponding eigenfunc- tions. Clearly show the calculations and state the reasoning justifying your conclusions.
7. Let the 2 x 2 matrix A have real, distinct eigenvalues and μ. Suppose that an eigenvector of A is (1,0)" and an eigenvector of u is (-1,1)". Sketch the phase portraits of is x = Ax for the following cases: (1) 0 0. وچستان ستان ج

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Consider the problem: y″ + 2y′ + (1 − \)y = 0, y'(0) = y' (π) = 0 1. Find the eigenvalues and eigenfunctions. 2. Verify that the eigenfunctions of the problem are orthogonal on [0,7] to the weight function w(x) = −1. 3. Find an orthonormnal set of eigenfunction on [0, 7].