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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

Solutions for Chapter 10.1
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Textbook: Differential Equations
Edition: 4
Author: Stephen W. Goode
ISBN: 9780321964670

Chapter 10.1: Definition of the Laplace Transform includes 38 full step-by-step solutions. Since 38 problems in chapter 10.1: Definition of the Laplace Transform have been answered, more than 20041 students have viewed full step-by-step 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.

Key Math Terms and definitions covered in this textbook
  • Cayley-Hamilton 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. Block-diagonal: 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, off-diagonal 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, ... , Aj-Ib. 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·

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