 6.3.1: In each of 1 through 6, sketch the graph of the given function on t...
 6.3.2: In each of 1 through 6, sketch the graph of the given function on t...
 6.3.3: In each of 1 through 6, sketch the graph of the given function on t...
 6.3.4: In each of 1 through 6, sketch the graph of the given function on t...
 6.3.5: In each of 1 through 6, sketch the graph of the given function on t...
 6.3.6: In each of 1 through 6, sketch the graph of the given function on t...
 6.3.7: In each of 7 through 12:(a) Sketch the graph of the given function....
 6.3.8: In each of 7 through 12:(a) Sketch the graph of the given function....
 6.3.9: In each of 7 through 12:(a) Sketch the graph of the given function....
 6.3.10: In each of 7 through 12:(a) Sketch the graph of the given function....
 6.3.11: In each of 7 through 12:(a) Sketch the graph of the given function....
 6.3.12: In each of 7 through 12:(a) Sketch the graph of the given function....
 6.3.13: In each of 13 through 18, find the Laplace transform of the given f...
 6.3.14: In each of 13 through 18, find the Laplace transform of the given f...
 6.3.15: In each of 13 through 18, find the Laplace transform of the given f...
 6.3.16: In each of 13 through 18, find the Laplace transform of the given f...
 6.3.17: In each of 13 through 18, find the Laplace transform of the given f...
 6.3.18: In each of 13 through 18, find the Laplace transform of the given f...
 6.3.19: In each of 19 through 24, find the inverse Laplace transform of the...
 6.3.20: In each of 19 through 24, find the inverse Laplace transform of the...
 6.3.21: In each of 19 through 24, find the inverse Laplace transform of the...
 6.3.22: In each of 19 through 24, find the inverse Laplace transform of the...
 6.3.23: In each of 19 through 24, find the inverse Laplace transform of the...
 6.3.24: In each of 19 through 24, find the inverse Laplace transform of the...
 6.3.25: Suppose that F(s) = L{f(t)} exists for s > a 0.(a) Show that if c i...
 6.3.26: In each of 26 through 29, use the results of to find the inverse La...
 6.3.27: In each of 26 through 29, use the results of to find the inverse La...
 6.3.28: In each of 26 through 29, use the results of to find the inverse La...
 6.3.29: In each of 26 through 29, use the results of to find the inverse La...
 6.3.30: In each of 30 through 33, find the Laplace transform of the given f...
 6.3.31: In each of 30 through 33, find the Laplace transform of the given f...
 6.3.32: In each of 30 through 33, find the Laplace transform of the given f...
 6.3.33: In each of 30 through 33, find the Laplace transform of the given f...
 6.3.34: Let f satisfy f(t + T) = f(t) for all t 0 and for some fixed positi...
 6.3.35: In each of 35 through 38, use the result of to find the Laplace tra...
 6.3.36: In each of 35 through 38, use the result of to find the Laplace tra...
 6.3.37: In each of 35 through 38, use the result of to find the Laplace tra...
 6.3.38: In each of 35 through 38, use the result of to find the Laplace tra...
 6.3.39: (a) If f(t) = 1 u1(t), find L{f(t)}; compare with 30. Sketch the gr...
 6.3.40: . Consider the function p defined byp(t) =t, 0 t < 1,2 t, 1 t < 2; ...
Solutions for Chapter 6.3: Step Functions
Full solutions for Elementary Differential Equations  10th Edition
ISBN: 9780470458327
Solutions for Chapter 6.3: Step Functions
Get Full SolutionsThis textbook survival guide was created for the textbook: Elementary Differential Equations, edition: 10. Elementary Differential Equations was written by and is associated to the ISBN: 9780470458327. Chapter 6.3: Step Functions includes 40 full stepbystep solutions. Since 40 problems in chapter 6.3: Step Functions have been answered, more than 12107 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

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

Column space C (A) =
space of all combinations of the columns of A.

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

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

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

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Solvable system Ax = b.
The right side b is in the column space of A.

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.

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.