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Solutions for Chapter 5: Modeling with Higher-Order Differential Equations

Differential Equations with Boundary-Value Problems, | 8th Edition | ISBN: 9781111827069 | Authors: Dennis G. Zill, Warren S. Wright

Full solutions for Differential Equations with Boundary-Value Problems, | 8th Edition

ISBN: 9781111827069

Differential Equations with Boundary-Value Problems, | 8th Edition | ISBN: 9781111827069 | Authors: Dennis G. Zill, Warren S. Wright

Solutions for Chapter 5: Modeling with Higher-Order Differential Equations

Solutions for Chapter 5
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Textbook: Differential Equations with Boundary-Value Problems,
Edition: 8
Author: Dennis G. Zill, Warren S. Wright
ISBN: 9781111827069

This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Differential Equations with Boundary-Value Problems,, edition: 8. Since 28 problems in chapter 5: Modeling with Higher-Order Differential Equations have been answered, more than 20275 students have viewed full step-by-step solutions from this chapter. Chapter 5: Modeling with Higher-Order Differential Equations includes 28 full step-by-step solutions. Differential Equations with Boundary-Value Problems, was written by and is associated to the ISBN: 9781111827069.

Key Math Terms and definitions covered in this textbook
  • Cofactor Cij.

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

  • 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).

  • 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.

  • Full column rank r = n.

    Independent columns, N(A) = {O}, no free variables.

  • 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.

  • Hankel matrix H.

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

  • Hilbert matrix hilb(n).

    Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.

  • 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.

  • Kirchhoff's Laws.

    Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

  • 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.

  • Linear transformation T.

    Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

  • Normal equation AT Ax = ATb.

    Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.

  • Orthogonal matrix Q.

    Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

  • Partial pivoting.

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

  • Similar matrices A and B.

    Every B = M-I AM has the same eigenvalues as A.

  • 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).

  • Transpose matrix AT.

    Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.

  • 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·

  • Volume of box.

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

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