 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 11800 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.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

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

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

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!).

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

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

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 •

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

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.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

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.

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·

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.