 11.3.11.1.47: In 110 determine whether the function is even, odd, or neither.f(x)...
 11.3.11.1.48: In 110 determine whether the function is even, odd, or neither.f(x)...
 11.3.11.1.49: In 110 determine whether the function is even, odd, or neither.f(x)...
 11.3.11.1.50: In 110 determine whether the function is even, odd, or neither.f(x)...
 11.3.11.1.51: In 110 determine whether the function is even, odd, or neither.f(x)...
 11.3.11.1.52: In 110 determine whether the function is even, odd, or neither.f(x)...
 11.3.11.1.53: In 110 determine whether the function is even, odd, or neither.f(x)...
 11.3.11.1.54: In 110 determine whether the function is even, odd, or neither.f(x)...
 11.3.11.1.55: In 110 determine whether the function is even, odd, or neither.f(x)...
 11.3.11.1.56: In 110 determine whether the function is even, odd, or neither.f(x) x5
 11.3.11.1.57: In 1124 expand the given function in an appropriate cosine or sine ...
 11.3.11.1.58: In 1124 expand the given function in an appropriate cosine or sine ...
 11.3.11.1.59: In 1124 expand the given function in an appropriate cosine or sine ...
 11.3.11.1.60: In 1124 expand the given function in an appropriate cosine or sine ...
 11.3.11.1.61: In 1124 expand the given function in an appropriate cosine or sine ...
 11.3.11.1.62: In 1124 expand the given function in an appropriate cosine or sine ...
 11.3.11.1.63: In 1124 expand the given function in an appropriate cosine or sine ...
 11.3.11.1.64: In 1124 expand the given function in an appropriate cosine or sine ...
 11.3.11.1.65: In 1124 expand the given function in an appropriate cosine or sine ...
 11.3.11.1.66: In 1124 expand the given function in an appropriate cosine or sine ...
 11.3.11.1.67: In 1124 expand the given function in an appropriate cosine or sine ...
 11.3.11.1.68: In 1124 expand the given function in an appropriate cosine or sine ...
 11.3.11.1.69: In 1124 expand the given function in an appropriate cosine or sine ...
 11.3.11.1.70: In 1124 expand the given function in an appropriate cosine or sine ...
 11.3.11.1.71: In 2534 find the halfrange cosine and sine expansions of the given...
 11.3.11.1.72: In 2534 find the halfrange cosine and sine expansions of the given...
 11.3.11.1.73: In 2534 find the halfrange cosine and sine expansions of the given...
 11.3.11.1.74: In 2534 find the halfrange cosine and sine expansions of the given...
 11.3.11.1.75: In 2534 find the halfrange cosine and sine expansions of the given...
 11.3.11.1.76: In 2534 find the halfrange cosine and sine expansions of the given...
 11.3.11.1.77: In 2534 find the halfrange cosine and sine expansions of the given...
 11.3.11.1.78: In 2534 find the halfrange cosine and sine expansions of the given...
 11.3.11.1.79: In 2534 find the halfrange cosine and sine expansions of the given...
 11.3.11.1.80: In 2534 find the halfrange cosine and sine expansions of the given...
 11.3.11.1.81: In 3538 expand the given function in a Fourier series. f(x) x2, 0 x 2p
 11.3.11.1.82: In 3538 expand the given function in a Fourier series. f(x) x, 0 x p
 11.3.11.1.83: In 3538 expand the given function in a Fourier series.f(x) x 1, 0 x 1
 11.3.11.1.84: In 3538 expand the given function in a Fourier series. f(x) 2 x, 0 x 2
 11.3.11.1.85: In 39 and 40 proceed as in Example 4 to find a particular solution ...
 11.3.11.1.86: In 39 and 40 proceed as in Example 4 to find a particular solution ...
 11.3.11.1.87: In 41 and 42 proceed as in Example 4 to find a particular solution ...
 11.3.11.1.88: In 41 and 42 proceed as in Example 4 to find a particular solution ...
 11.3.11.1.89: (a) Solve the differential equation in 39, x 10x f(t), subject to t...
 11.3.11.1.90: (a) Solve the differential equation in 41, subject to the initial c...
 11.3.11.1.91: Suppose a uniform beam of length L is simply supported at x 0 and a...
 11.3.11.1.92: Proceed as in to find a particular solution yp(x) when the load per...
 11.3.11.1.93: When a uniform beam is supported by an elastic foundation and subje...
 11.3.11.1.94: Prove properties (a), (c), (d), (f), and (g) in Theorem 11.3.1.
 11.3.11.1.95: There is only one function that is both even and odd. What is it?
 11.3.11.1.96: As we know from Chapter 4, the general solution of the differential...
 11.3.11.1.97: In 51 and 52 use a CAS to plot graphs of partial sums {SN(x)} of th...
 11.3.11.1.98: In 51 and 52 use a CAS to plot graphs of partial sums {SN(x)} of th...
 11.3.11.1.99: Is your answer in or in unique? Give a function f defined on a symm...
Solutions for Chapter 11.3: Fourier Series
Full solutions for Differential Equations with BoundaryValue Problems,  8th Edition
ISBN: 9781111827069
Solutions for Chapter 11.3: Fourier Series
Get Full SolutionsChapter 11.3: Fourier Series includes 53 full stepbystep solutions. Since 53 problems in chapter 11.3: Fourier Series have been answered, more than 21200 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Differential Equations with BoundaryValue Problems,, edition: 8. This expansive textbook survival guide covers the following chapters and their solutions. Differential Equations with BoundaryValue Problems, was written by and is associated to the ISBN: 9781111827069.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

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.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

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

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

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

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

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.

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.