 3.3.1: In each of 1 through 6 use Eulers formula to write the given expres...
 3.3.2: In each of 1 through 6 use Eulers formula to write the given expres...
 3.3.3: In each of 1 through 6 use Eulers formula to write the given expres...
 3.3.4: In each of 1 through 6 use Eulers formula to write the given expres...
 3.3.5: In each of 1 through 6 use Eulers formula to write the given expres...
 3.3.6: In each of 1 through 6 use Eulers formula to write the given expres...
 3.3.7: In each of 7 through 16 find the general solution of the given diff...
 3.3.8: In each of 7 through 16 find the general solution of the given diff...
 3.3.9: In each of 7 through 16 find the general solution of the given diff...
 3.3.10: In each of 7 through 16 find the general solution of the given diff...
 3.3.11: In each of 7 through 16 find the general solution of the given diff...
 3.3.12: In each of 7 through 16 find the general solution of the given diff...
 3.3.13: In each of 7 through 16 find the general solution of the given diff...
 3.3.14: In each of 7 through 16 find the general solution of the given diff...
 3.3.15: In each of 7 through 16 find the general solution of the given diff...
 3.3.16: In each of 7 through 16 find the general solution of the given diff...
 3.3.17: In each of 17 through 22 find the solution of the given initial val...
 3.3.18: In each of 17 through 22 find the solution of the given initial val...
 3.3.19: In each of 17 through 22 find the solution of the given initial val...
 3.3.20: In each of 17 through 22 find the solution of the given initial val...
 3.3.21: In each of 17 through 22 find the solution of the given initial val...
 3.3.22: In each of 17 through 22 find the solution of the given initial val...
 3.3.23: Consider the initial value problem 3u u + 2u = 0, u(0) = 2, u (0) =...
 3.3.24: Consider the initial value problem 5u + 2u + 7u = 0, u(0) = 2, u (0...
 3.3.25: Consider the initial value problem y + 2y + 6y = 0, y(0) = 2, y (0)...
 3.3.26: Consider the initial value problem y + 2ay + (a2 + 1)y = 0, y(0) = ...
 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 that cost = (eit + eit)/2, sin t = (eit ...
 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 that d dt ert = rert for any comp...
 3.3.32: Let the realvalued functions p and q be continuous on the open int...
 3.3.33: If the functions y1 and y2 are a fundamental set of solutions of y ...
 3.3.34: Change of Variables. Sometimes a differential equation with variabl...
 3.3.35: In each of 35 through 42 use the method of to solve the given equat...
 3.3.36: In each of 35 through 42 use the method of to solve the given equat...
 3.3.37: In each of 35 through 42 use the method of to solve the given equat...
 3.3.38: In each of 35 through 42 use the method of to solve the given equat...
 3.3.39: In each of 35 through 42 use the method of to solve the given equat...
 3.3.40: In each of 35 through 42 use the method of to solve the given equat...
 3.3.41: In each of 35 through 42 use the method of to solve the given equat...
 3.3.42: In each of 35 through 42 use the method of to solve the given equat...
 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 o...
 3.3.45: In each of 44 through 46 try to transform the given equation into o...
 3.3.46: In each of 44 through 46 try to transform the given equation into o...
Solutions for Chapter 3.3: Complex Roots of the Characteristic Equation
Full solutions for Elementary Differential Equations and Boundary Value Problems  9th Edition
ISBN: 9780470383346
Solutions for Chapter 3.3: Complex Roots of the Characteristic Equation
Get Full SolutionsElementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470383346. Chapter 3.3: Complex Roots of the Characteristic Equation includes 46 full stepbystep solutions. This 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 12652 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: 9.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

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

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.

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.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

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

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

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

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

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