 4.2.1: In each of 1 through 6, express the given complex number in the for...
 4.2.2: In each of 1 through 6, express the given complex number in the for...
 4.2.3: In each of 1 through 6, express the given complex number in the for...
 4.2.4: In each of 1 through 6, express the given complex number in the for...
 4.2.5: In each of 1 through 6, express the given complex number in the for...
 4.2.6: In each of 1 through 6, express the given complex number in the for...
 4.2.7: In each of 7 through 10, follow the procedure illustrated in Exampl...
 4.2.8: In each of 7 through 10, follow the procedure illustrated in Exampl...
 4.2.9: In each of 7 through 10, follow the procedure illustrated in Exampl...
 4.2.10: In each of 7 through 10, follow the procedure illustrated in Exampl...
 4.2.11: In each of 11 through 28, find the general solution of the given di...
 4.2.12: In each of 11 through 28, find the general solution of the given di...
 4.2.13: In each of 11 through 28, find the general solution of the given di...
 4.2.14: In each of 11 through 28, find the general solution of the given di...
 4.2.15: In each of 11 through 28, find the general solution of the given di...
 4.2.16: In each of 11 through 28, find the general solution of the given di...
 4.2.17: In each of 11 through 28, find the general solution of the given di...
 4.2.18: In each of 11 through 28, find the general solution of the given di...
 4.2.19: In each of 11 through 28, find the general solution of the given di...
 4.2.20: In each of 11 through 28, find the general solution of the given di...
 4.2.21: In each of 11 through 28, find the general solution of the given di...
 4.2.22: In each of 11 through 28, find the general solution of the given di...
 4.2.23: In each of 11 through 28, find the general solution of the given di...
 4.2.24: In each of 11 through 28, find the general solution of the given di...
 4.2.25: In each of 11 through 28, find the general solution of the given di...
 4.2.26: In each of 11 through 28, find the general solution of the given di...
 4.2.27: In each of 11 through 28, find the general solution of the given di...
 4.2.28: In each of 11 through 28, find the general solution of the given di...
 4.2.29: In each of 29 through 36, find the solution of the given initial va...
 4.2.30: In each of 29 through 36, find the solution of the given initial va...
 4.2.31: In each of 29 through 36, find the solution of the given initial va...
 4.2.32: In each of 29 through 36, find the solution of the given initial va...
 4.2.33: In each of 29 through 36, find the solution of the given initial va...
 4.2.34: In each of 29 through 36, find the solution of the given initial va...
 4.2.35: In each of 29 through 36, find the solution of the given initial va...
 4.2.36: In each of 29 through 36, find the solution of the given initial va...
 4.2.37: Show that the general solution of y(4) y = 0 can be written asy = c...
 4.2.38: Consider the equation y(4) y = 0.(a) Use Abels formula [ 20(d) of S...
 4.2.39: Consider the springmass system, shown in Figure 4.2.4, consisting o...
 4.2.40: In this problem we outline one way to show that if r1, ... ,rn are ...
 4.2.41: In this problem we indicate one way to show that if r = r1 is a roo...
Solutions for Chapter 4.2: Homogeneous Equations with Constant Coefficients
Full solutions for Elementary Differential Equations and Boundary Value Problems  10th Edition
ISBN: 9780470458310
Solutions for Chapter 4.2: Homogeneous Equations with Constant Coefficients
Get Full SolutionsChapter 4.2: Homogeneous Equations with Constant Coefficients includes 41 full stepbystep solutions. Since 41 problems in chapter 4.2: Homogeneous Equations with Constant Coefficients have been answered, more than 16865 students have viewed full stepbystep solutions from this chapter. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470458310. This 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.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

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.

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.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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

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.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

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.

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

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

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.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

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