 11.3.1: In each of 1 through 5, solve the given problem by means of an eige...
 11.3.2: In each of 1 through 5, solve the given problem by means of an eige...
 11.3.3: In each of 1 through 5, solve the given problem by means of an eige...
 11.3.4: In each of 1 through 5, solve the given problem by means of an eige...
 11.3.5: In each of 1 through 5, solve the given problem by means of an eige...
 11.3.6: In each of 6 through 9, determine a formal eigenfunction series exp...
 11.3.7: In each of 6 through 9, determine a formal eigenfunction series exp...
 11.3.8: In each of 6 through 9, determine a formal eigenfunction series exp...
 11.3.9: In each of 6 through 9, determine a formal eigenfunction series exp...
 11.3.10: In each of 10 through 13, determine whether there is any value of t...
 11.3.11: In each of 10 through 13, determine whether there is any value of t...
 11.3.12: In each of 10 through 13, determine whether there is any value of t...
 11.3.13: In each of 10 through 13, determine whether there is any value of t...
 11.3.14: Let 1, 2, ... , n, ... be the normalized eigenfunctions of the diff...
 11.3.15: Let L be a second order linear differential operator. Show that the...
 11.3.16: Show that the problemy+ 2y = 2x, y(0) = 1, y(1) = 0has the solution...
 11.3.17: Consider the problemy+ p(x)y+ q(x)y = 0, y(0) = a, y(1) = b.Let y =...
 11.3.18: Using the method of 17, transform the problemy+ 2y = 2 4x, y(0) = 1...
 11.3.19: In each of 19 through 22, use eigenfunction expansions to find the ...
 11.3.20: In each of 19 through 22, use eigenfunction expansions to find the ...
 11.3.21: In each of 19 through 22, use eigenfunction expansions to find the ...
 11.3.22: In each of 19 through 22, use eigenfunction expansions to find the ...
 11.3.23: Consider the boundary value problemr(x)ut = [p(x)ux]x q(x)u + F(x),...
 11.3.24: In each of 24 and 25, use the method indicated in to solve the give...
 11.3.25: In each of 24 and 25, use the method indicated in to solve the give...
 11.3.26: The method of eigenfunction expansions is often useful for nonhomog...
 11.3.27: In this problem we explore a little further the analogy between Stu...
 11.3.28: 28 through 36 indicate a way of solving nonhomogeneous boundary val...
 11.3.29: 28 through 36 indicate a way of solving nonhomogeneous boundary val...
 11.3.30: 28 through 36 indicate a way of solving nonhomogeneous boundary val...
 11.3.31: 28 through 36 indicate a way of solving nonhomogeneous boundary val...
 11.3.32: 28 through 36 indicate a way of solving nonhomogeneous boundary val...
 11.3.33: 28 through 36 indicate a way of solving nonhomogeneous boundary val...
 11.3.34: 28 through 36 indicate a way of solving nonhomogeneous boundary val...
 11.3.35: 28 through 36 indicate a way of solving nonhomogeneous boundary val...
 11.3.36: 28 through 36 indicate a way of solving nonhomogeneous boundary val...
Solutions for Chapter 11.3: Nonhomogeneous Boundary Value Problems
Full solutions for Elementary Differential Equations and Boundary Value Problems  10th Edition
ISBN: 9780470458310
Solutions for Chapter 11.3: Nonhomogeneous Boundary Value Problems
Get Full SolutionsChapter 11.3: Nonhomogeneous Boundary Value Problems includes 36 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 10. Since 36 problems in chapter 11.3: Nonhomogeneous Boundary Value Problems have been answered, more than 16819 students have viewed full stepbystep solutions from this chapter. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470458310.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Outer product uv T
= column times row = rank one matrix.

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.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

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