 6.6.1: Establish the commutative, distributive, and associative properties...
 6.6.2: Find an example different from the one in the text showing that (f ...
 6.6.3: Show, by means of the example f(t) = sin t, that f f is not necessa...
 6.6.4: In each of 4 through 7 find the Laplace transform of the given func...
 6.6.5: In each of 4 through 7 find the Laplace transform of the given func...
 6.6.6: In each of 4 through 7 find the Laplace transform of the given func...
 6.6.7: In each of 4 through 7 find the Laplace transform of the given func...
 6.6.8: In each of 8 through 11 find the inverse Laplace transform of the g...
 6.6.9: In each of 8 through 11 find the inverse Laplace transform of the g...
 6.6.10: In each of 8 through 11 find the inverse Laplace transform of the g...
 6.6.11: In each of 8 through 11 find the inverse Laplace transform of the g...
 6.6.12: In each of 8 through 11 find the inverse Laplace transform of the g...
 6.6.13: In each of 13 through 20 express the solution of the given initial ...
 6.6.14: In each of 13 through 20 express the solution of the given initial ...
 6.6.15: In each of 13 through 20 express the solution of the given initial ...
 6.6.16: In each of 13 through 20 express the solution of the given initial ...
 6.6.17: In each of 13 through 20 express the solution of the given initial ...
 6.6.18: In each of 13 through 20 express the solution of the given initial ...
 6.6.19: In each of 13 through 20 express the solution of the given initial ...
 6.6.20: In each of 13 through 20 express the solution of the given initial ...
 6.6.21: Consider the equation (t) + t 0 k(t )() d = f(t), in which f and k ...
 6.6.22: Consider the Volterra integral equation (see 21) (t) + t 0 (t )() d...
 6.6.23: Consider the Volterra integral equation (see 21) (t) + t 0 (t )() d...
 6.6.24: Consider the Volterra integral equation (see 21) (t) + t 0 (t )() d...
 6.6.25: Consider the Volterra integral equation (see 21) (t) + t 0 (t )() d...
 6.6.26: There are also equations, known as integrodifferential equations, ...
 6.6.27: There are also equations, known as integrodifferential equations, ...
 6.6.28: There are also equations, known as integrodifferential equations, ...
 6.6.29: The Tautochrone. A problem of interest in the history of mathematic...
Solutions for Chapter 6.6: The Convolution Integral
Full solutions for Elementary Differential Equations and Boundary Value Problems  9th Edition
ISBN: 9780470383346
Solutions for Chapter 6.6: The Convolution Integral
Get Full SolutionsSince 29 problems in chapter 6.6: The Convolution Integral have been answered, more than 12779 students have viewed full stepbystep solutions from this chapter. Chapter 6.6: The Convolution Integral includes 29 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 9. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470383346. 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.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

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

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

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.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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).

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 columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

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

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.

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

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

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