 6.6.1: . Establish the commutative, distributive, and associative properti...
 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 fun...
 6.6.5: In each of 4 through 7, find the Laplace transform of the given fun...
 6.6.6: In each of 4 through 7, find the Laplace transform of the given fun...
 6.6.7: In each of 4 through 7, find the Laplace transform of the given fun...
 6.6.8: In each of 8 through 11, find the inverse Laplace transform of the ...
 6.6.9: In each of 8 through 11, find the inverse Laplace transform of the ...
 6.6.10: In each of 8 through 11, find the inverse Laplace transform of the ...
 6.6.11: In each of 8 through 11, find the inverse Laplace transform of the ...
 6.6.12: (a) If f(t) = tm and g(t) = tn, where m and n are positive integers...
 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) + t0k(t )() d = f(t),in which f and k are ...
 6.6.22: Consider the Volterra integral equation (see 21)(t) + t0(t )() d = ...
 6.6.23: In each of 23 through 25:(a) Solve the given Volterra integral equa...
 6.6.24: In each of 23 through 25:(a) Solve the given Volterra integral equa...
 6.6.25: In each of 23 through 25:(a) Solve the given Volterra integral equa...
 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  10th Edition
ISBN: 9780470458327
Solutions for Chapter 6.6: The Convolution Integral
Get Full SolutionsThis textbook survival guide was created for the textbook: Elementary Differential Equations, edition: 10. Chapter 6.6: The Convolution Integral includes 29 full stepbystep solutions. Since 29 problems in chapter 6.6: The Convolution Integral have been answered, more than 11407 students have viewed full stepbystep solutions from this chapter. Elementary Differential Equations was written by and is associated to the ISBN: 9780470458327. This expansive textbook survival guide covers the following chapters and their solutions.

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.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Iterative method.
A sequence of steps intended to approach the desired solution.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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