 7.1.1: Apply the definition in (1) to find directly the Laplace transforms...
 7.1.2: Apply the definition in (1) to find directly the Laplace transforms...
 7.1.3: Apply the definition in (1) to find directly the Laplace transforms...
 7.1.4: Apply the definition in (1) to find directly the Laplace transforms...
 7.1.5: Apply the definition in (1) to find directly the Laplace transforms...
 7.1.6: Apply the definition in (1) to find directly the Laplace transforms...
 7.1.7: Apply the definition in (1) to find directly the Laplace transforms...
 7.1.8: Apply the definition in (1) to find directly the Laplace transforms...
 7.1.9: Apply the definition in (1) to find directly the Laplace transforms...
 7.1.10: Apply the definition in (1) to find directly the Laplace transforms...
 7.1.11: Use the transforms in Fig. 7.1.2 to find the Laplace transforms of ...
 7.1.12: Use the transforms in Fig. 7.1.2 to find the Laplace transforms of ...
 7.1.13: Use the transforms in Fig. 7.1.2 to find the Laplace transforms of ...
 7.1.14: Use the transforms in Fig. 7.1.2 to find the Laplace transforms of ...
 7.1.15: Use the transforms in Fig. 7.1.2 to find the Laplace transforms of ...
 7.1.16: Use the transforms in Fig. 7.1.2 to find the Laplace transforms of ...
 7.1.17: Use the transforms in Fig. 7.1.2 to find the Laplace transforms of ...
 7.1.18: Use the transforms in Fig. 7.1.2 to find the Laplace transforms of ...
 7.1.19: Use the transforms in Fig. 7.1.2 to find the Laplace transforms of ...
 7.1.20: Use the transforms in Fig. 7.1.2 to find the Laplace transforms of ...
 7.1.21: Use the transforms in Fig. 7.1.2 to find the Laplace transforms of ...
 7.1.22: Use the transforms in Fig. 7.1.2 to find the Laplace transforms of ...
 7.1.23: Use the transforms in Fig. 7.1.2 to find the inverse Laplace transf...
 7.1.24: Use the transforms in Fig. 7.1.2 to find the inverse Laplace transf...
 7.1.25: Use the transforms in Fig. 7.1.2 to find the inverse Laplace transf...
 7.1.26: Use the transforms in Fig. 7.1.2 to find the inverse Laplace transf...
 7.1.27: Use the transforms in Fig. 7.1.2 to find the inverse Laplace transf...
 7.1.28: Use the transforms in Fig. 7.1.2 to find the inverse Laplace transf...
 7.1.29: Use the transforms in Fig. 7.1.2 to find the inverse Laplace transf...
 7.1.30: Use the transforms in Fig. 7.1.2 to find the inverse Laplace transf...
 7.1.31: Use the transforms in Fig. 7.1.2 to find the inverse Laplace transf...
 7.1.32: Use the transforms in Fig. 7.1.2 to find the inverse Laplace transf...
 7.1.33: Derive the transform of f .t / D sin kt by the method used in the t...
 7.1.34: Derive the transform of f .t / D sinh kt by the method used in the ...
 7.1.35: Use the tabulated integral Z eax cos bx dx D eax a2 C b2 .a cos bx ...
 7.1.36: Show that the function f .t / D sin.et2 / is of exponential order a...
 7.1.37: Given a>0, let f .t / D 1 if 0 5 tf .t / D 0 if t = a.First, sketch...
 7.1.38: Given that 0<a<b, let f .t / D 1 if a 5 t<b, f .t / D 0 if either t...
 7.1.39: The unit staircase function is defined as follows: f .t / D n if n ...
 7.1.40: (a) The graph of the function f is shown in Fig. 7.1.10. Show that ...
 7.1.41: The graph of the squarewave function g.t / is shown in Fig. 7.1.11...
 7.1.42: Given constants a and b, define h.t / for t = 0 by h.t / D ( a if n...
Solutions for Chapter 7.1: Laplace Transforms and Inverse Transforms
Full solutions for Differential Equations and Boundary Value Problems: Computing and Modeling  5th Edition
ISBN: 9780321796981
Solutions for Chapter 7.1: Laplace Transforms and Inverse Transforms
Get Full SolutionsChapter 7.1: Laplace Transforms and Inverse Transforms includes 42 full stepbystep solutions. Since 42 problems in chapter 7.1: Laplace Transforms and Inverse Transforms have been answered, more than 38630 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Differential Equations and Boundary Value Problems: Computing and Modeling was written by and is associated to the ISBN: 9780321796981. This textbook survival guide was created for the textbook: Differential Equations and Boundary Value Problems: Computing and Modeling, edition: 5.

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.

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).

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

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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.

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

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.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

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

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

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

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

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

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.