 2.10.1: Use Properties 1 and 2 to find the Laplace transform of each of the...
 2.10.2: Use Properties 1 and 2 to find the Laplace transform of each of the...
 2.10.3: Use Properties 1 and 2 to find the Laplace transform of each of the...
 2.10.4: Use Properties 1 and 2 to find the Laplace transform of each of the...
 2.10.5: t5I2 (see Exercise 9, Section 2.9)
 2.10.6: Let F(s) = I? { f (t)), and suppose that f (t)/t has a limit as t a...
 2.10.7: Use Equation (*) of to find the Laplace transform of each of the fo...
 2.10.8: Find the inverse Laplace transform of each of the following functio...
 2.10.9: Find the inverse Laplace transform of each of the following functio...
 2.10.10: Find the inverse Laplace transform of each of the following functio...
 2.10.11: Find the inverse Laplace transform of each of the following functio...
 2.10.12: Find the inverse Laplace transform of each of the following functio...
 2.10.13: Find the inverse Laplace transform of each of the following functio...
 2.10.14: Find the inverse Laplace transform of each of the following functio...
 2.10.15: Find the inverse Laplace transform of each of the following functio...
 2.10.16: Find the inverse Laplace transform of each of the following functio...
 2.10.17: Let F (s) = e { f (t)). Show that Thus, if we know how to invert F'...
 2.10.18: Use the result of to invert each of the following Laplace transform...
 2.10.19: Solve each of the following initialvalue problems by the method of...
 2.10.20: Solve each of the following initialvalue problems by the method of...
 2.10.21: Solve each of the following initialvalue problems by the method of...
 2.10.22: Solve each of the following initialvalue problems by the method of...
 2.10.23: Solve each of the following initialvalue problems by the method of...
 2.10.24: Solve each of the following initialvalue problems by the method of...
Solutions for Chapter 2.10: Some useful properties of Laplace transforms
Full solutions for Differential Equations and Their Applications: An Introduction to Applied Mathematics  3rd Edition
ISBN: 9780387908069
Solutions for Chapter 2.10: Some useful properties of Laplace transforms
Get Full SolutionsDifferential Equations and Their Applications: An Introduction to Applied Mathematics was written by and is associated to the ISBN: 9780387908069. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Differential Equations and Their Applications: An Introduction to Applied Mathematics, edition: 3. Chapter 2.10: Some useful properties of Laplace transforms includes 24 full stepbystep solutions. Since 24 problems in chapter 2.10: Some useful properties of Laplace transforms have been answered, more than 6101 students have viewed full stepbystep solutions from this chapter.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

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

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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

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

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

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

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

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

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.

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.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

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.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

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 space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.