(a) Find all positive values of λ for which the following boundary value problem has a...

(a) Find all positive values of λ for which the following boundary value problem has a nonzero solution. What are the corresponding eigenfunctions? X′′ + 4Xʹ + (λ + 4) X = 0, X′(0) = 0 and X′(1) = 0. Hint: the roots of its auxiliary equation are –2 ± σi, where λ = σ².

(b) Is λ = 0 an eigenvalue of this boundary value problem? Why or why not?

Solutions

Expert Solution

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

Find the values of λ (eigenvalues) for which the given problem has a nontrivial solution. Also...

Find the values of λ (eigenvalues) for which the given problem has a nontrivial solution. Also determine the corresponding nontrivial solutions (eigenfunctions). y''+2λy=0; 0<x<π, y(0)=0, y'(π)=0

Find the eigenvalues and eigenfunctions of the given boundary value problem. Assume that all eigenvalues are...

Find the eigenvalues and eigenfunctions of the given boundary value problem. Assume that all eigenvalues are real. (Let n represent an arbitrary positive number.) y''+λy= 0， y（0）= 0， y'（π）= 0

Find the eigenfunctions for the following boundary value problem. x2y?? ? 15xy? + (64 + ?)?y...

Find the eigenfunctions for the following boundary value problem. x2y?? ? 15xy? + (64 + ?)?y ?=? 0, y(e?1) ?=? 0, ?y(1) ?=? 0. In the eigenfunction take the arbitrary constant (either c1 or c2) from the general solution to be 1

For which values of λ does the system of equations (λ − 2)x + y =...

For which values of λ does the system of equations (λ − 2)x + y = 0 x + (λ − 2)y = 0 have nontrivial solutions? (That is, solutions other than x = y = 0.) For each such λ find a nontrivial solution.

Use the method of Undetermined Coefficients to find the solution of the boundary value problem x^2...

Use the method of Undetermined Coefficients to find the solution of the boundary value problem x^2 y '' + y' + 2y = 6x +2 y(1) = 0 y(2) = 1

Find eigenvalue (?) and eigenfunction and evaluate orthogonality from the given boundary value problem. ?2?′′ +...

Find eigenvalue (?) and eigenfunction and evaluate orthogonality from the given boundary value problem. ?2?′′ + ??′ + ?? = 0, ?(1) = 0, ?(5) = 0. Hint: Use Cauchy-Euler Equation, (textbook pp141-143).

Write a program to solve the boundary value problem ? ′′ = ? ′ + 2?...

Write a program to solve the boundary value problem ? ′′ = ? ′ + 2? + cos ? for ? ? [0, ?/2] with ?( 0) = 0.3, ?( ?/ 2) = 0.1. Check your numerical solution with actual using necessary plot.(MATLAB)

Find the eigenvalues λn and eigenfunctions yn(x) for the given boundary-value problem. (Give your answers in...

Find the eigenvalues λn and eigenfunctions yn(x) for the given boundary-value problem. (Give your answers in terms of k, making sure that each value of k corresponds to two unique eigenvalues.) y'' + λy = 0, y(−π) = 0, y(π) = 0 λ2k − 1 =, k=1,2,3,... y2k − 1(x) =, k=1,2,3,... λ2k =, k=1,2,3,... y2k(x) =, k=1,2,3,...

Find the eigenvalues λn and eigenfunctions yn(x) for the given boundary-value problem. (Give your answers in...

Find the eigenvalues λn 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.) y'' + λy = 0, y(0) = 0, y(π/6) = 0 λn = , n = 1, 2, 3, yn(x) = , n = 1, 2, 3,

Find the eigenvalues λn and eigenfunctions yn(x) for the given boundary-value problem. (Give your answers in...

Subjects