 3.2.1: In 1 through 6, show directly that the given functions are linearly...
 3.2.2: In 1 through 6, show directly that the given functions are linearly...
 3.2.3: In 1 through 6, show directly that the given functions are linearly...
 3.2.4: In 1 through 6, show directly that the given functions are linearly...
 3.2.5: In 1 through 6, show directly that the given functions are linearly...
 3.2.6: In 1 through 6, show directly that the given functions are linearly...
 3.2.7: In 7 through 12, use the Wronskian to prove that the given function...
 3.2.8: In 7 through 12, use the Wronskian to prove that the given function...
 3.2.9: In 7 through 12, use the Wronskian to prove that the given function...
 3.2.10: In 7 through 12, use the Wronskian to prove that the given function...
 3.2.11: In 7 through 12, use the Wronskian to prove that the given function...
 3.2.12: In 7 through 12, use the Wronskian to prove that the given function...
 3.2.13: In 13 through 20, a thirdorder homogeneous linear equation and thr...
 3.2.14: In 13 through 20, a thirdorder homogeneous linear equation and thr...
 3.2.15: In 13 through 20, a thirdorder homogeneous linear equation and thr...
 3.2.16: In 13 through 20, a thirdorder homogeneous linear equation and thr...
 3.2.17: In 13 through 20, a thirdorder homogeneous linear equation and thr...
 3.2.18: In 13 through 20, a thirdorder homogeneous linear equation and thr...
 3.2.19: In 13 through 20, a thirdorder homogeneous linear equation and thr...
 3.2.20: In 13 through 20, a thirdorder homogeneous linear equation and thr...
 3.2.21: In 21 through 24, a nonhomogeneous differential equation, a complem...
 3.2.22: In 21 through 24, a nonhomogeneous differential equation, a complem...
 3.2.23: In 21 through 24, a nonhomogeneous differential equation, a complem...
 3.2.24: In 21 through 24, a nonhomogeneous differential equation, a complem...
 3.2.25: Let Ly D y00 C py0 C qy. Suppose that y1 and y2 are two functions s...
 3.2.26: (a) Find by inspection particular solutions of the two nonhomogeneo...
 3.2.27: Prove directly that the functions f1.x/ 1; f2.x/ D x; and f3.x/ D x...
 3.2.28: Generalize the method of to prove directly that the functions f0.x/...
 3.2.29: Use the result of and the definition of linear independence to prov...
 3.2.30: Verify that y1 D x and y2 D x2 are linearly independent solutions o...
 3.2.31: This problem indicates why we can impose only n initial conditions ...
 3.2.32: Prove that an nthorder homogeneous linear differential equation sa...
 3.2.33: Suppose that the three numbers r1, r2, and r3 are distinct. Show th...
 3.2.34: Assume as known that the Vandermonde determinant V D 1 1 1 r1 r2 rn...
 3.2.35: According to of Section 3.1, the Wronskian W .y1; y2/ of two soluti...
 3.2.36: Suppose that one solution y1.x/ of the homogeneous secondorder lin...
 3.2.37: Before applying Eq. (19) with a given homogeneous secondorder line...
 3.2.38: In each of 38 through 42, a differential equation and one solution ...
 3.2.39: In each of 38 through 42, a differential equation and one solution ...
 3.2.40: In each of 38 through 42, a differential equation and one solution ...
 3.2.41: In each of 38 through 42, a differential equation and one solution ...
 3.2.42: In each of 38 through 42, a differential equation and one solution ...
 3.2.43: First note that y1.x/ D x is one solution of Legendres equation of ...
 3.2.44: First verify by substitution that y1.x/ D x1=2 cos x is one solutio...
Solutions for Chapter 3.2: General Solutions of Linear Equations
Full solutions for Differential Equations and Boundary Value Problems: Computing and Modeling  5th Edition
ISBN: 9780321796981
Solutions for Chapter 3.2: General Solutions of Linear Equations
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Differential Equations and Boundary Value Problems: Computing and Modeling was written by and is associated to the ISBN: 9780321796981. Chapter 3.2: General Solutions of Linear Equations includes 44 full stepbystep solutions. Since 44 problems in chapter 3.2: General Solutions of Linear Equations have been answered, more than 16675 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Differential Equations and Boundary Value Problems: Computing and Modeling, edition: 5.

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!

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

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.

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.

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

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.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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.

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

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

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

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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.

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).

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Stiffness matrix
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.