- 9.3.1: In 1 through 10, a function f .t / defined on an interval 0f .t / D...
- 9.3.2: In 1 through 10, a function f .t / defined on an interval 1f .t / D...
- 9.3.3: In 1 through 10, a function f .t / defined on an interval 2f .t / D...
- 9.3.4: In 1 through 10, a function f .t / defined on an interval 3f .t / D...
- 9.3.5: In 1 through 10, a function f .t / defined on an interval 4f .t / D...
- 9.3.6: In 1 through 10, a function f .t / defined on an interval 5f .t / D...
- 9.3.7: In 1 through 10, a function f .t / defined on an interval 6f .t / D...
- 9.3.8: In 1 through 10, a function f .t / defined on an interval 7f .t / D...
- 9.3.9: In 1 through 10, a function f .t / defined on an interval 8f .t / D...
- 9.3.10: In 1 through 10, a function f .t / defined on an interval 9f .t / D...
- 9.3.11: Find formal Fourier series solutions of the endpoint value problems...
- 9.3.12: Find formal Fourier series solutions of the endpoint value problems...
- 9.3.13: Find formal Fourier series solutions of the endpoint value problems...
- 9.3.14: Find formal Fourier series solutions of the endpoint value problems...
- 9.3.15: Find a formal Fourier series solution of the endpoint value problem...
- 9.3.16: (a) Derive the solution x.t / D t .sin 2t /=.sin 2/ of the endpoint...
- 9.3.17: (a) Suppose that f is an even function. Show that Z 0 a f .t / dt D...
- 9.3.18: By Example 2 of Section 9.2, the Fourier series of the period 2 fun...
- 9.3.19: Begin with the Fourier series t D 2 X1 nD1 .1/nC1 n sin nt; < t < ;...
- 9.3.20: Substitute t D =2 and t D in the series of to obtain the summations...
- 9.3.21: Substitute t D =2 and t D in the series of to obtain the summations...
- 9.3.22: (Odd half-multiple cosine series) Let f .t / be given for 0
- 9.3.23: Given: f .t / D t, 0
- 9.3.24: Given the endpoint value problem x00 x D t; x.0/ D 0; x0 ./ D 0; no...
- 9.3.25: In this problem we outline the proof of Theorem 2. Suppose that f ....
Solutions for Chapter 9.3: Fourier Sine and Cosine Series
Full solutions for Differential Equations and Boundary Value Problems: Computing and Modeling | 5th Edition
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
Upper triangular systems are solved in reverse order Xn to Xl.
peA) = det(A - AI) has peA) = zero matrix.
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
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.
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.
Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and
Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.
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.
Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).