 7.4.1: Prove the generalization of Theorem 7.4.1, as expressed in the sent...
 7.4.2: In this problem we outline a proof of Theorem 7.4.3 in the case n =...
 7.4.3: Show that theWronskians of two fundamental sets of solutions of the...
 7.4.4: . If x1 = y and x2 = y, then the second order equationy + p(t)y + q...
 7.4.5: Show that the general solution of x = P(t)x + g(t)is the sum of any...
 7.4.6: Consider the vectors x(1)(t) =t1and x(2)(t) =t22t.(a) Compute the W...
 7.4.7: Consider the vectors x(1)(t) =t22tand x(2)(t) =etet, and answer the...
 7.4.8: Let x(1), ... , x(m) be solutions of x = P(t)x on the interval < t ...
 7.4.9: Let x(1), ... , x(n) be linearly independent solutions of x = P(t)x...
Solutions for Chapter 7.4: Elementary Differential Equations, 10th Edition 9780470458327 William E. Boyce / Richard C. DiPrima
Full solutions for Elementary Differential Equations  10th Edition
ISBN: 9780470458327
Solutions for Chapter 7.4: Elementary Differential Equations, 10th Edition 9780470458327 William E. Boyce / Richard C. DiPrima
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 9 problems in chapter 7.4: Elementary Differential Equations, 10th Edition 9780470458327 William E. Boyce / Richard C. DiPrima have been answered, more than 12270 students have viewed full stepbystep solutions from this chapter. Elementary Differential Equations was written by and is associated to the ISBN: 9780470458327. Chapter 7.4: Elementary Differential Equations, 10th Edition 9780470458327 William E. Boyce / Richard C. DiPrima includes 9 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Differential Equations, edition: 10.

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!

Column space C (A) =
space of all combinations of the columns of A.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib IIĀ· Condition numbers measure the sensitivity of the output to change in the input.

Cyclic shift
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.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

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

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

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

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.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

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.

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