 9.4.1: Find the steady periodic solution xsp.t / of each of the differenti...
 9.4.2: Find the steady periodic solution xsp.t / of each of the differenti...
 9.4.3: Find the steady periodic solution xsp.t / of each of the differenti...
 9.4.4: Find the steady periodic solution xsp.t / of each of the differenti...
 9.4.5: Find the steady periodic solution xsp.t / of each of the differenti...
 9.4.6: Find the steady periodic solution xsp.t / of each of the differenti...
 9.4.7: m D 1, k D 9; F .t / is the odd function of period 2with F .t / D 1...
 9.4.8: m D 2, k D 10; F .t / is the odd function of period 2 with F .t / D...
 9.4.9: m D 3, k D 12; F .t / is the odd function of period 2with F .t / D ...
 9.4.10: m D 1, k D 42; F .t / is the odd function of period 2 with F .t / D...
 9.4.11: m D 3, k D 48; F .t / is the even function of period 2with F .t / D...
 9.4.12: m D 2, k D 50; F .t / is the odd function of period 2with F .t / D ...
 9.4.13: m D 1, c D 0:1, k D 4; F .t / is the force of 1.
 9.4.14: m D 2, c D 0:1, k D 18; F .t / is the force of 3.
 9.4.15: m D 3, c D 1, k D 30; F .t / is the force of 5.
 9.4.16: m D 1, c D 0:01, k D 4; F .t / is the force of 4.
 9.4.17: Consider a forced damped massandspring system with m D 1 4 slug, ...
 9.4.18: Consider a forced damped massandspring system with m D 1, c D 0:0...
 9.4.19: Suppose the functions f .t / and g.t / are periodic with periods P ...
 9.4.20: If p=q is irrational, prove that the function f .t / D cos pt C cos...
Solutions for Chapter 9.4: Applications of Fourier Series
Full solutions for Differential Equations and Boundary Value Problems: Computing and Modeling  5th Edition
ISBN: 9780321796981
Solutions for Chapter 9.4: Applications of Fourier Series
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CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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.

Iterative method.
A sequence of steps intended to approach the desired solution.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Solvable system Ax = b.
The right side b is in the column space of A.

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

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.