[answered] Ma 221 Homework 9 Due: Nov 17, 2016 Name (Printed): Recitat

please finish and complete the entire worksheet and show work for each question.

Ma 221 Homework 9 Due: Nov 17, 2016 Name (Printed): Recitation: Pledge and Sign: General Instructions: Write up solutions to the following set of questions and submit in recitation on

 

the date indicated. Please staple this cover sheet to your solution pages.

 

Legibility, organization of the solution, and clearly stated reasoning where appropriate are all important.

 

Points will be deducted for disorganized work or insufficient explanations.

 

Collaboration with classmates is acceptable and encouraged but all students must write up the solutions on

 

their own. Collaborators (up to groups of three) should be identified on the top of the front page. All

 

submitted work must be pledged and signed. 1. Nonhomogeneous Boundary Value Problem - solution not unique.

 

Consider the boundary value problem, y 00 + ? 2 y = g(x) on 0 < x < 2 with boundary

 

conditions, y(0) = 0 and y(2) = 0.

 

(a) Verify that the homogeneous boundary value problem has a one-parameter family of

 

nontrivial solutions, y = C sin(?x).

 

(b) Show that the nonhomogeneous BVP has no solution for the case, g(x) = x.

 

(c) Show that the nonhomogeneous BVP has infinitely many solutions for g(x) = ?x2 + 2x. 2. Eigenvalue Problem including first derivative in linear operator.

 

Find the eigenvalues and eigenfunctions for the boundary value problem,

 

y 00 + 2y 0 + ?y = 0 on 0 < x < ?, y 0 (0) = 0, y 0 (?) = 0. 3. Eigenvalue Problem for Cauchy-Euler Equation.

 

Find the eigenvalues and eigenfunctions for the boundary value problem,

 

x2 y 00 + xy 0 + ?y = 0, y(1) = 0, y(2) = 0.

 

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