Consider the ODE u" + lambda u=0 with the boundary conditions u'(0)=u'(M)=0, where M is a...

Consider the ODE u" + lambda u=0 with the boundary conditions u'(0)=u'(M)=0, where M is a fixed positive constant. So u=0 is a solution for every lambda,

Determine the eigen values of the differential operators: that is

a: find all lambda such that the above ODE with boundary conditions has non trivial sol.

b. And, what are the non trivial eigenvalues you obtain for each eigenvalue

Solutions

Expert Solution

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

Consider the following wave equation for u(t, x) with boundary and initial conditions defined for 0...

Consider the following wave equation for u(t, x) with boundary and initial conditions defined for 0 ≤ x ≤ 2 and t ≥ 0. ∂ 2u ∂t2 = 0.01 ∂ 2u ∂x2 (0 ≤ x ≤ 2, t ≥ 0) (1) ∂u ∂x(t, 0) = 0 and ∂u ∂x(t, 2) = 0 (2) u(0, x) = f(x) = x if 0 ≤ x ≤ 1 1 if 1 ≤ x ≤ 2. (a) Compute the coefficients a0, a1, a2, ....

For the wave equation, utt = c2uxx, with the following boundary and initial conditions, u(x, 0)...

For the wave equation, utt = c2uxx, with the following boundary and initial conditions, u(x, 0) = 0 ut(x, 0) = 0.1x(π − x) u(0,t) = u(π,t) = 0 (a) Solve the problem using the separation of variables. (b) Solve the problem using D’Alembert’s solution. Hint: I would suggest doing an odd expansion of ut(x,0) first; the final solution should be exactly like the one in (a).

Consider the 1D wave equation d^2u/dt^2 = c^2( d^2u/dx^2) with the following boundary conditions: u(0, t)...

Consider the 1D wave equation d^2u/dt^2 = c^2( d^2u/dx^2) with the following boundary conditions: u(0, t) = ux (L, t) = 0 . (a) Use separation of variables technique to calculate the eigenvalues, eigenfunctions and general solution. (b) Now, assume L = π and c = 1. With initial conditions u(x, 0) = 0 and ut(x, 0) = 1, calculate the solution for u(x, t). (c) With initial conditions u(x, 0) = sin(x/2) and ut(x, 0) = 2 sin(x/2) −...

u'' + sinu = sinx (-1<x<1) u(-1)=1, u'(1)=0 solve this boundary value problem.

Solve the Boundary Value Problem, PDE: Utt-a2uxx=0, 0≤x≤1, 0≤t<∞ BCs: u(0,t)=0 u(1,t)=cos(t) u(x,0)=0 ut(x,0)=0

Find the linear space of eigenfunctions for the problem with periodic boundary conditions u′′(x) = λu(x)...

Find the linear space of eigenfunctions for the problem with periodic boundary conditions u′′(x) = λu(x) u(0) = u(2π) u′(0) = u′(2π) for (a) λ = −1 (b) λ = 0 (c) λ = 1. Note that you should look for nontrivial eigenfunctions

6. Consider the one dimensional wave equation with boundary conditions and initial conditions: PDE : utt...

6. Consider the one dimensional wave equation with boundary conditions and initial conditions: PDE : utt = c 2 uxx, BC : u(0, t) = u(L, t) = 0, IC : u(x, 0) = f(x), ut(x, 0) = g(x) a) Suppose c = 1, L = 1, f(x) = 180x 2 (1 − x), and g(x) = 0. Using the first 10 terms in the series, plot the solution surface and enough time snapshots to display the dynamics of the...

Consider the boundary value problem X ′′ +λX=0 , X ′ (0)=0 , X′(π)=0 . Find...

Consider the boundary value problem X ′′ +λX=0 , X ′ (0)=0 , X′(π)=0 . Find all real values of λ for which there is a non-trivial solution of the problem and find the corresponding solution.

Given a second order ode m y’’ + c y’ + k y = 0 with m, c...

Given a second order ode m y’’ + c y’ + k y = 0 with m, c and k all positive. (like a mass‐spring system with damping) Argue that the solution will always be damped; the exponential portion can never be positive regardless of the particular m, c and k.

(PDE) Find the series soln to Ut=Uxx on -2<x<2, T>0 with Dirichlet boundary { U(t,-2)=0=U(t, 2)...

(PDE) Find the series soln to Ut=Uxx on -2<x<2, T>0 with Dirichlet boundary { U(t,-2)=0=U(t, 2) initial condition { U(0,x) = { x, IxI <1

Subjects