 10.4.1: In each of 1 through 6, determine whether the given function is eve...
 10.4.2: In each of 1 through 6, determine whether the given function is eve...
 10.4.3: In each of 1 through 6, determine whether the given function is eve...
 10.4.4: In each of 1 through 6, determine whether the given function is eve...
 10.4.5: In each of 1 through 6, determine whether the given function is eve...
 10.4.6: In each of 1 through 6, determine whether the given function is eve...
 10.4.7: In each of 7 through 12, a function f is given on an interval of le...
 10.4.8: In each of 7 through 12, a function f is given on an interval of le...
 10.4.9: In each of 7 through 12, a function f is given on an interval of le...
 10.4.10: In each of 7 through 12, a function f is given on an interval of le...
 10.4.11: In each of 7 through 12, a function f is given on an interval of le...
 10.4.12: In each of 7 through 12, a function f is given on an interval of le...
 10.4.13: Prove that any function can be expressed as the sum of two other fu...
 10.4.14: Find the coefficients in the cosine and sine series described in Ex...
 10.4.15: In each of 15 through 22:(a) Find the required Fourier series for t...
 10.4.16: In each of 15 through 22:(a) Find the required Fourier series for t...
 10.4.17: In each of 15 through 22:(a) Find the required Fourier series for t...
 10.4.18: In each of 15 through 22:(a) Find the required Fourier series for t...
 10.4.19: In each of 15 through 22:(a) Find the required Fourier series for t...
 10.4.20: In each of 15 through 22:(a) Find the required Fourier series for t...
 10.4.21: In each of 15 through 22:(a) Find the required Fourier series for t...
 10.4.22: In each of 15 through 22:(a) Find the required Fourier series for t...
 10.4.23: In each of 23 through 26:(a) Find the required Fourier series for t...
 10.4.24: In each of 23 through 26:(a) Find the required Fourier series for t...
 10.4.25: In each of 23 through 26:(a) Find the required Fourier series for t...
 10.4.26: In each of 23 through 26:(a) Find the required Fourier series for t...
 10.4.27: In each of 27 through 30, a function is given on an interval 0 < x ...
 10.4.28: In each of 27 through 30, a function is given on an interval 0 < x ...
 10.4.29: In each of 27 through 30, a function is given on an interval 0 < x ...
 10.4.30: In each of 27 through 30, a function is given on an interval 0 < x ...
 10.4.31: Prove that if f is an odd function, thenLLf(x) dx = 0.
 10.4.32: Prove properties 2 and 3 of even and odd functions, as stated in th...
 10.4.33: Prove that the derivative of an even function is odd and that the d...
 10.4.34: Let F(x) = x 0 f(t) dt. Show that if f is even, then F is odd, and ...
 10.4.35: From the Fourier series for the square wave in Example 1 of Section...
 10.4.36: From the Fourier series for the triangular wave (Example 1 of Secti...
 10.4.37: Assume that f has a Fourier sine seriesf(x) = n=1bn sin(nx/L), 0 x ...
 10.4.38: Let f be extended into (L, 2L] in an arbitrary manner subject to th...
 10.4.39: Let f be extended into (L, 2L] in an arbitrary manner subject to th...
 10.4.40: (a) How should f, originally defined on [0, L], be extended so as t...
Solutions for Chapter 10.4: Even and Odd Functions
Full solutions for Elementary Differential Equations and Boundary Value Problems  10th Edition
ISBN: 9780470458310
Solutions for Chapter 10.4: Even and Odd Functions
Get Full SolutionsThis textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 10. This expansive textbook survival guide covers the following chapters and their solutions. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470458310. Chapter 10.4: Even and Odd Functions includes 40 full stepbystep solutions. Since 40 problems in chapter 10.4: Even and Odd Functions have been answered, more than 18249 students have viewed full stepbystep solutions from this chapter.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

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

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

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.

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.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

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.

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

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.