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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

Solutions for Chapter 7.1
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ISBN: 9780321796981

Chapter 7.1: Laplace Transforms and Inverse Transforms includes 42 full step-by-step solutions. Since 42 problems in chapter 7.1: Laplace Transforms and Inverse Transforms have been answered, more than 38630 students have viewed full step-by-step 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.

Key Math Terms and definitions covered in this textbook
• 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 n-l - 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.

• lA-II = 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 e-iO , 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)j-I and det V = product of (Xk - Xi) for k > i.