The eigenfunctions for y" + 2y' + dy = y(0) = 1, y(1) = 0 are {e* sin(nræ)_1O {e2r sin(프) 1( {e-"sin(푸)0 od {e-2n sin(nTz)F_1이 {e"'sin(프)EO
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- Show that A=[0110] has no real eigenvalues.What is the resulting differential equation after the initial substitution from the appropriate 2nd order transformation? Do not transpose terms to the other side of the equation. Continue from the solution obtained with the appropriate 2nd order transformation. If x = 4, C1 = -1/2, and C2 = 0, what is one possible value of y?What is the resulting differential equation after the initial substitution from the appropriate 2nd order transformation? Do not transpose terms to the other side of the equation. Continue from the solution obtained with the appropriate 2nd order transformation. If x = 3, C1 = 0, and C2 = 0, what is the value of y?
- Consider the eigenvalue problem y"+2y'+(lambda)y=0; y(0)=0, y'(1)=0. Show that the eigenvalues are all positive and that the nth positive eigenvalue is (lambda)n= an^2+1 with associated eigenfunction yn(x)= e^-x sin(anx) where anx is the nth positive root of tanz=zA is a 3 X 3matrix with eigenvectors v1 ,v2 and v3 corresponding to eigenvalues λ1 = -1/3,λ2=1/3 and λ3 = 1, respectively, and x .Find A kx. What happens as k becomes large (i.e., k--->∞ )What is the resulting differential equation after the initial substitution from the appropriate 2nd order transformation? Do not transpose terms to the other side of the equation.
- The real part of the eigenvalues is (negative, positive, zero) so the equilibrium is (a center, an unstable spiral, a stable spiral)Suppose that the damped harmonic oscil-lator is governed by the second order linear differential equationI try to use the eigenvalue to find eigenvector, and I've already found the relation between x and y. I can substitute x=1 and get y=-i, but the solution provided is substituting x=-1 and get y=i. There is a problem, how can I know which value of x should I choose?
- How can I show that a twice continuously differentiable function with a lipschitz continuous hessian with all eigenvalues≥mu is mu strongly convex?5.Consider the eigenvalues given by equation (39). Show that (σ1X+σ2Y)2−4(σ1σ2−α1α2)XY=(σ1X−σ2Y)2+4α1α2XY.σ1X+σ2Y2−4σ1σ2−α1α2XY=σ1X−σ2Y2+4α1α2XY. Hence conclude that the eigenvalues can never be complex-valued.Determine if the spherical harmonics (Yl,m) are eigenfunctions of the L3 operator. If yes, find their eigenvalues. SHOW FULL AND COMPLETE PROCEDURE IN A CLEAR AND ORDERED WAY.