 3.3.1: In each of 1 through 6, use Eulers formula to write the given expre...
 3.3.2: In each of 1 through 6, use Eulers formula to write the given expre...
 3.3.3: In each of 1 through 6, use Eulers formula to write the given expre...
 3.3.4: In each of 1 through 6, use Eulers formula to write the given expre...
 3.3.5: In each of 1 through 6, use Eulers formula to write the given expre...
 3.3.6: In each of 1 through 6, use Eulers formula to write the given expre...
 3.3.7: In each of 7 through 16, find the general solution of the given dif...
 3.3.8: In each of 7 through 16, find the general solution of the given dif...
 3.3.9: In each of 7 through 16, find the general solution of the given dif...
 3.3.10: In each of 7 through 16, find the general solution of the given dif...
 3.3.11: In each of 7 through 16, find the general solution of the given dif...
 3.3.12: In each of 7 through 16, find the general solution of the given dif...
 3.3.13: In each of 7 through 16, find the general solution of the given dif...
 3.3.14: In each of 7 through 16, find the general solution of the given dif...
 3.3.15: In each of 7 through 16, find the general solution of the given dif...
 3.3.16: In each of 7 through 16, find the general solution of the given dif...
 3.3.17: In each of 17 through 22, find the solution of the given initial va...
 3.3.18: In each of 17 through 22, find the solution of the given initial va...
 3.3.19: In each of 17 through 22, find the solution of the given initial va...
 3.3.20: In each of 17 through 22, find the solution of the given initial va...
 3.3.21: In each of 17 through 22, find the solution of the given initial va...
 3.3.22: In each of 17 through 22, find the solution of the given initial va...
 3.3.23: Consider the initial value problem3u u+ 2u = 0, u(0) = 2, u(0) = 0....
 3.3.24: Consider the initial value problem5u+ 2u+ 7u = 0, u(0) = 2, u(0) = ...
 3.3.25: Consider the initial value problemy+ 2y+ 6y = 0, y(0) = 2, y(0) = 0...
 3.3.26: Consider the initial value problemy+ 2ay+ (a2 + 1)y = 0, y(0) = 1, ...
 3.3.27: Show that W(et cost, et sint) = e2t .
 3.3.28: In this problem we outline a different derivation of Eulers formula...
 3.3.29: Using Eulers formula, show thatcost = (eit + eit)/2, sin t = (eit e...
 3.3.30: If ert is given by Eq. (13), show that e(r1+r2)t = er1t er2t for an...
 3.3.31: If ert is given by Eq. (13), show thatddt ert = rertfor any complex...
 3.3.32: Consider the differential equationay+ by+ cy = 0,where b2 4ac < 0 a...
 3.3.33: If the functions y1 and y2 are a fundamental set of solutions of y+...
 3.3.34: Euler Equations. An equation of the formt2 d2ydt2 + tdydt + y = 0, ...
 3.3.35: In each of 35 through 42, use the method of to solve the given equa...
 3.3.36: In each of 35 through 42, use the method of to solve the given equa...
 3.3.37: In each of 35 through 42, use the method of to solve the given equa...
 3.3.38: In each of 35 through 42, use the method of to solve the given equa...
 3.3.39: In each of 35 through 42, use the method of to solve the given equa...
 3.3.40: In each of 35 through 42, use the method of to solve the given equa...
 3.3.41: In each of 35 through 42, use the method of to solve the given equa...
 3.3.42: In each of 35 through 42, use the method of to solve the given equa...
 3.3.43: In this problem we determine conditions on p and q that enable Eq. ...
 3.3.44: In each of 44 through 46, try to transform the given equation into ...
 3.3.45: In each of 44 through 46, try to transform the given equation into ...
 3.3.46: In each of 44 through 46, 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  10th Edition
ISBN: 9780470458310
Solutions for Chapter 3.3: Complex Roots of the Characteristic Equation
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 46 problems in chapter 3.3: Complex Roots of the Characteristic Equation have been answered, more than 16300 students have viewed full stepbystep solutions from this chapter. Chapter 3.3: Complex Roots of the Characteristic Equation includes 46 full stepbystep solutions. 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.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

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

Covariance matrix:E.
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.

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.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

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.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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.

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

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

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 ().

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·