 5.5.5.5.1: A SturmLiouville eigenvalue problem is called selfadjoint if p u d...
 5.5.5.5.2: Prove that the eigenfunctions corresponding to different eigenvalue...
 5.5.5.5.3: Consider the eigenvalue problem L() = (x), subject to a given set o...
 5.5.5.5.4: Give an example of an eigenvalue problem with more than one eigenfu...
 5.5.5.5.5: Consider L = d2 dx2 + 6 d dx + 9. (a) Show that L(erx)=(r + 3)2erx....
 5.5.5.5.6: Consider L = d2 dx2 + 6 d dx + 9. (a) Show that L(erx)=(r + 3)2erx....
 5.5.5.5.7: For L = d dx p d dx + q with p and q real, carefully show that L() ...
 5.5.5.5.8: Consider a fourthorder linear differential operator, L = d4 dx4 . ...
 5.5.5.5.9: For the eigenvalue problem d4 dx4 + ex = 0 subject to the boundary ...
 5.5.5.5.10: For the eigenvalue problem d4 dx4 + ex = 0 subject to the boundary ...
 5.5.5.5.11: *(a) Suppose that L = p(x) d2 dx2 + r(x) d dx + q(x). Consider b a ...
 5.5.5.5.12: Consider nonselfadjoint operators as in Exercise 5.5.11. The eigen...
 5.5.5.5.13: Using the result of Exercise 5.5.11, prove the following part of th...
 5.5.5.5.14: Using the result of Exercise 5.5.11, prove the following part of th...
 5.5.5.5.15: Consider the eigenvalue problem d dr (r d dr ) + r = 0 (0 <r< 1), s...
 5.5.5.5.16: Consider the onedimensional wave equation for nonconstant density ...
 5.5.5.5.17: Consider the onedimensional wave equation with c constant, 2u t2 =...
 5.5.5.5.18: Prove Greens formula for the SturmLiouville operator L(y) = d dx (p...
 5.5.5.5A.1: Prove that the eigenvalues of real symmetric matrices are real
 5.5.5.5A.2: (a) Show that the matrix A = 1 0 2 1 has only one independent eigen...
 5.5.5.5A.3: Consider the eigenvectors of the matrix A = 6 4 1 3 . (a) Show that...
 5.5.5.5A.4: Solve dv/dt = Av using matrix methods if *(a) A = 6 2 2 3 , v(0) = ...
 5.5.5.5A.5: Show that the eigenvalues are real and the eigenvectors orthogonal:...
 5.5.5.5A.6: For a matrix A whose entries are complex numbers, the complex conju...
Solutions for Chapter 5.5: SturmLiouville Eigenvalue Problems
Full solutions for Applied Partial Differential Equations with Fourier Series and Boundary Value Problems  5th Edition
ISBN: 9780321797056
Solutions for Chapter 5.5: SturmLiouville Eigenvalue Problems
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 5.5: SturmLiouville Eigenvalue Problems includes 24 full stepbystep solutions. Since 24 problems in chapter 5.5: SturmLiouville Eigenvalue Problems have been answered, more than 8049 students have viewed full stepbystep solutions from this chapter. Applied Partial Differential Equations with Fourier Series and Boundary Value Problems was written by and is associated to the ISBN: 9780321797056. This textbook survival guide was created for the textbook: Applied Partial Differential Equations with Fourier Series and Boundary Value Problems, edition: 5.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.