 2.13.1: Compute the convolution of each of the following pairs of functions...
 2.13.2: Compute the convolution of each of the following pairs of functions...
 2.13.3: Compute the convolution of each of the following pairs of functions...
 2.13.4: Compute the convolution of each of the following pairs of functions...
 2.13.5: Compute the convolution of each of the following pairs of functions...
 2.13.6: Compute the convolution of each of the following pairs of functions...
 2.13.7: Use Theorem 9 to invert each of the following Laplace transforms.
 2.13.8: Use Theorem 9 to invert each of the following Laplace transforms.
 2.13.9: Use Theorem 9 to invert each of the following Laplace transforms.
 2.13.10: Use Theorem 9 to invert each of the following Laplace transforms.
 2.13.11: Use Theorem 9 to invert each of the following Laplace transforms.
 2.13.12: Use Theorem 9 to invert each of the following Laplace transforms.
 2.13.13: Use Theorem 9 to find the solution y(t) of each of the following in...
 2.13.14: Use Theorem 9 to find the solution y(t) of each of the following in...
 2.13.15: Use Theorem 9 to find the solution y(t) of each of the following in...
 2.13.16: Use Theorem 9 to find the solution y(t) of each of the following in...
 2.13.17: Use Theorem 9 to find the solution y(t) of each of the following in...
 2.13.18: Use Theorem 9 to find the solution y(t) of each of the following in...
 2.13.19: Use Theorem 9 to find the solution y(t) of each of the following in...
 2.13.20: Use Theorem 9 to find the solution y(t) of each of the following in...
Solutions for Chapter 2.13: The convolution integral
Full solutions for Differential Equations and Their Applications: An Introduction to Applied Mathematics  3rd Edition
ISBN: 9780387908069
Solutions for Chapter 2.13: The convolution integral
Get Full SolutionsDifferential Equations and Their Applications: An Introduction to Applied Mathematics was written by and is associated to the ISBN: 9780387908069. This textbook survival guide was created for the textbook: Differential Equations and Their Applications: An Introduction to Applied Mathematics, edition: 3. Chapter 2.13: The convolution integral includes 20 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 20 problems in chapter 2.13: The convolution integral have been answered, more than 6088 students have viewed full stepbystep solutions from this chapter.

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

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

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

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

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

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

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

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

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

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.