 7.2.7.1.57: In 130 use appropriate algebra and Theorem 7.2.1 to find the given ...
 7.2.7.1.58: In 130 use appropriate algebra and Theorem 7.2.1 to find the given ...
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 7.2.7.1.87: In 3140 use the Laplace transform to solve the given initialvalue ...
 7.2.7.1.88: In 3140 use the Laplace transform to solve the given initialvalue ...
 7.2.7.1.89: In 3140 use the Laplace transform to solve the given initialvalue ...
 7.2.7.1.90: In 3140 use the Laplace transform to solve the given initialvalue ...
 7.2.7.1.91: In 3140 use the Laplace transform to solve the given initialvalue ...
 7.2.7.1.92: In 3140 use the Laplace transform to solve the given initialvalue ...
 7.2.7.1.93: In 3140 use the Laplace transform to solve the given initialvalue ...
 7.2.7.1.94: In 3140 use the Laplace transform to solve the given initialvalue ...
 7.2.7.1.95: In 3140 use the Laplace transform to solve the given initialvalue ...
 7.2.7.1.96: In 3140 use the Laplace transform to solve the given initialvalue ...
 7.2.7.1.97: The inverse forms of the results in in Exercises 7.1 are In 41 and ...
 7.2.7.1.98: The inverse forms of the results in in Exercises 7.1 are In 41 and ...
 7.2.7.1.99: (a) With a slight change in notation the transform in (6) is the sa...
 7.2.7.1.100: Make up two functions f1 and f2 that have the same Laplace transfor...
 7.2.7.1.101: Reread (iii) in the Remarks on page 288. Find the zeroinput and th...
 7.2.7.1.102: Suppose f(t) is a function for which f(t) is piecewise continuous a...
Solutions for Chapter 7.2: The Laplace Transform
Full solutions for Differential Equations with BoundaryValue Problems,  8th Edition
ISBN: 9781111827069
Solutions for Chapter 7.2: The Laplace Transform
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 46 problems in chapter 7.2: The Laplace Transform have been answered, more than 21207 students have viewed full stepbystep solutions from this chapter. Chapter 7.2: The Laplace Transform includes 46 full stepbystep solutions. Differential Equations with BoundaryValue Problems, was written by and is associated to the ISBN: 9781111827069. This textbook survival guide was created for the textbook: Differential Equations with BoundaryValue Problems,, edition: 8.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

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

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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.

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.

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

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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.

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

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.