Consider the functional S[y] = ay(1)² + 1 dx ẞy², y(0) = 0, with a natural boundary condition at x = 1 and subject to the constraint 1 C[v] = vy(1)² + [* dx w(x) y² = 1, where a, ẞ and y are nonzero constants. Euler-Lagrange equation β d²y dx² +\w(x)y=0, y(0) = 0, (a — yλ) y(1) + ß y′(1) = 0, where is a Lagrange multiplier.

Consider the functional S[y] = ay(1)² + 1 dx ẞy², y(0) = 0, with a natural boundary condition at x = 1 and subject to the constraint 1 C[v] = vy(1)² + [* dx w(x) y² = 1, where a, ẞ and y are nonzero constants. Euler-Lagrange equation β d²y dx² +\w(x)y=0, y(0) = 0, (a — yλ) y(1) + ß y′(1) = 0, where is a Lagrange multiplier.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

a. Find the eigenvalues
Consider the system of differential equations For this system, the smaller eigenvalue is (-39/10) [Note-- you may want to view a phase plane plot (right click to open in a new window).] If y'= Ay is a differential equation, how would the solution curves behave? • All of the solutions curves would converge towards 0. (Stable node) All of the solution curves would run away from 0. (Unstable node) and the larger eigenvalue is The solution curves would race towards zero and then veer away towards infinity. (Saddle) The solution curves converge to different points. The solution to the above differential equation with initial values (0) = 4, y(0) = 3 is (t)= (49/12)*e^((-9/10)*t)-(1/12)*e^((-19/10)*t) y(t) = (49/12)*e^((-9/10)*t)+(5/12)*e^((-19/10)*t) da dt dy dt = -1.4x +0.75y, = 1.66667 3.4y. (-9/10)
Determine the eigenvalues and corresponding eigenfunctions for the following boundary value problem: Domain xe (0,7) X" + AX = 0 with boundary conditions: X(0)=X'(7)=0
Given that the matrix A has eigenvaluesX₁ = -5 with corresponding eigenvector ₁ = and A₂ = 3 with corresponding eigenvector 2 = A = 1 [:] , find A. [:]
Find the eigenvalues and eigenfunctions yn(x) for the given boundary-value problem. (Give your answers in terms of n, making sure that each value of n corresponds to a unique eigenvalue.) x²y" + xy' + 2y = 0, y(1)= 0, y(e) = 0 2n = Yn(x) = , n = 1, 2, 3, ... , n = 1, 2, 3,...
Find the eigenvalues ,, and eigenfunctions y(x) for the given boundary-value problem. (Give your answers in terms of n, making sure that each value of n corresponds to a unique eigenvalue.); y"+2y+ (+1)y= 0, y(0)= 0, y(4) = 0 (No Response) - 1, 2, 3, Y(x)(No Response) ne 1, 2, 3,...
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)