 10.1.1: For 112, use (10.1.1) to determine L[ f ].f (t) = t 1.
 10.1.2: For 112, use (10.1.1) to determine L[ f ].f (t) = e2t
 10.1.3: For 112, use (10.1.1) to determine L[ f ].f (t) = tet.
 10.1.4: For 112, use (10.1.1) to determine L[ f ].f (t) = sin bt, where b i...
 10.1.5: For 112, use (10.1.1) to determine L[ f ].f (t) = sinh bt, where b ...
 10.1.6: For 112, use (10.1.1) to determine L[ f ].f (t) = cosh bt, where b ...
 10.1.7: For 112, use (10.1.1) to determine L[ f ].
 10.1.8: For 112, use (10.1.1) to determine L[ f ].f (t) = 2t.
 10.1.9: For 112, use (10.1.1) to determine L[ f ].. f (t) =t2, 0 t 1,1, t > 1.
 10.1.10: For 112, use (10.1.1) to determine L[ f ].f (t) = 1, 0 t < 2,1, t 2.
 10.1.11: For 112, use (10.1.1) to determine L[ f ].f (t) = e2t cos 3t.
 10.1.12: For 112, use (10.1.1) to determine L[ f ].f (t) = et sin t.
 10.1.13: For 1322, use the linearity of L and the formulas derived in this s...
 10.1.14: For 1322, use the linearity of L and the formulas derived in this s...
 10.1.15: For 1322, use the linearity of L and the formulas derived in this s...
 10.1.16: For 1322, use the linearity of L and the formulas derived in this s...
 10.1.17: For 1322, use the linearity of L and the formulas derived in this s...
 10.1.18: For 1322, use the linearity of L and the formulas derived in this s...
 10.1.19: For 1322, use the linearity of L and the formulas derived in this s...
 10.1.20: For 1322, use the linearity of L and the formulas derived in this s...
 10.1.21: For 1322, use the linearity of L and the formulas derived in this s...
 10.1.22: For 1322, use the linearity of L and the formulas derived in this s...
 10.1.23: For 2330, sketch the given function and determine whether it is pie...
 10.1.24: For 2330, sketch the given function and determine whether it is pie...
 10.1.25: For 2330, sketch the given function and determine whether it is pie...
 10.1.26: For 2330, sketch the given function and determine whether it is pie...
 10.1.27: For 2330, sketch the given function and determine whether it is pie...
 10.1.28: For 2330, sketch the given function and determine whether it is pie...
 10.1.29: For 2330, sketch the given function and determine whether it is pie...
 10.1.30: For 2330, sketch the given function and determine whether it is pie...
 10.1.31: For 3134, sketch the given function and determine its Laplace trans...
 10.1.32: For 3134, sketch the given function and determine its Laplace trans...
 10.1.33: For 3134, sketch the given function and determine its Laplace trans...
 10.1.34: For 3134, sketch the given function and determine its Laplace trans...
 10.1.35: Recall that according to Eulers formula eibt = cos bt + i sin bt. S...
 10.1.36: Use the technique introduced in the previous problem to determine L...
 10.1.37: Use mathematical induction to prove that for every positive integer...
 10.1.38: (a) By making the change of variablest = x2 s (s > 0) in the integr...
Solutions for Chapter 10.1: Definition of the Laplace Transform
Full solutions for Differential Equations  4th Edition
ISBN: 9780321964670
Solutions for Chapter 10.1: Definition of the Laplace Transform
Get Full SolutionsChapter 10.1: Definition of the Laplace Transform includes 38 full stepbystep solutions. Since 38 problems in chapter 10.1: Definition of the Laplace Transform have been answered, more than 20041 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Differential Equations was written by and is associated to the ISBN: 9780321964670. This textbook survival guide was created for the textbook: Differential Equations, edition: 4.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Iterative method.
A sequence of steps intended to approach the desired solution.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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.

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

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

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

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

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

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

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·