 3.3.1: In each of 1 through 4, use Eulers formula to write the given expre...
 3.3.2: In each of 1 through 4, use Eulers formula to write the given expre...
 3.3.3: In each of 1 through 4, use Eulers formula to write the given expre...
 3.3.4: In each of 1 through 4, use Eulers formula to write the given expre...
 3.3.5: In each of 5 through 11, find the general solution of the given dif...
 3.3.6: In each of 5 through 11, find the general solution of the given dif...
 3.3.7: In each of 5 through 11, find the general solution of the given dif...
 3.3.8: In each of 5 through 11, find the general solution of the given dif...
 3.3.9: In each of 5 through 11, find the general solution of the given dif...
 3.3.10: In each of 5 through 11, find the general solution of the given dif...
 3.3.11: In each of 5 through 11, find the general solution of the given dif...
 3.3.12: In each of 12 through 15, find the solution of the given initial va...
 3.3.13: In each of 12 through 15, find the solution of the given initial va...
 3.3.14: In each of 12 through 15, find the solution of the given initial va...
 3.3.15: In each of 12 through 15, find the solution of the given initial va...
 3.3.16: Consider the initial value problem 3u__ u_ + 2u = 0, u(0) = 2, u_(0...
 3.3.17: Consider the initial value problem 5u__ + 2u_ + 7u = 0, u(0) = 2, u...
 3.3.18: Consider the initial value problem y__ + 2y_ + 6y = 0, y(0) = 2, y_...
 3.3.19: Show that W_et cos(t), et sin(t)_ = e2t .
 3.3.20: In this problem we outline a different derivation of Eulers formula...
 3.3.21: Using Eulers formula, show that eit + eit 2 = cos t, eit eit 2i = s...
 3.3.22: If ert is given by equation (14), show that e(r1+r2)t = er1t er2t f...
 3.3.23: Consider the differential equation ay__ + by_ + cy = 0, where b24ac...
 3.3.24: If the functions y1 and y2 are a fundamental set of solutions of y_...
 3.3.25: Euler Equations. An equation of the form t2 d2 y dt2 + t dy dt + y ...
 3.3.26: In each of 26 through 31, use the method of to solve the given equa...
 3.3.27: In each of 26 through 31, use the method of to solve the given equa...
 3.3.28: In each of 26 through 31, use the method of to solve the given equa...
 3.3.29: In each of 26 through 31, use the method of to solve the given equa...
 3.3.30: In each of 26 through 31, use the method of to solve the given equa...
 3.3.31: In each of 26 through 31, use the method of to solve the given equa...
 3.3.32: In this problem we determine conditions on p and q that enable equa...
 3.3.33: In each of 33 through 36, try to transform the given equation into ...
 3.3.34: In each of 33 through 36, try to transform the given equation into ...
 3.3.35: In each of 33 through 36, try to transform the given equation into ...
 3.3.36: In each of 33 through 36, try to transform the given equation into ...
Solutions for Chapter 3.3: Complex Roots of the Characteristic Equation
Full solutions for Elementary Differential Equations and Boundary Value Problems  11th Edition
ISBN: 9781119256007
Solutions for Chapter 3.3: Complex Roots of the Characteristic Equation
Get Full SolutionsChapter 3.3: Complex Roots of the Characteristic Equation includes 36 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 36 problems in chapter 3.3: Complex Roots of the Characteristic Equation have been answered, more than 12619 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 11. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9781119256007.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

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

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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!).

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

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

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.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

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

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

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