 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; p(...
Solutions for Chapter 6.3: Step Functions
Full solutions for Elementary Differential Equations and Boundary Value Problems  10th Edition
ISBN: 9780470458310
Solutions for Chapter 6.3: Step Functions
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 6.3: Step Functions includes 40 full stepbystep solutions. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470458310. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 10. Since 40 problems in chapter 6.3: Step Functions have been answered, more than 10528 students have viewed full stepbystep solutions from this chapter.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

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).

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(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.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.