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Solutions for Chapter 5.4: Euler Equations; Regular Singular Points

Elementary Differential Equations and Boundary Value Problems | 10th Edition | ISBN: 9780470458310 | Authors: William E. Boyce

Full solutions for Elementary Differential Equations and Boundary Value Problems | 10th Edition

ISBN: 9780470458310

Elementary Differential Equations and Boundary Value Problems | 10th Edition | ISBN: 9780470458310 | Authors: William E. Boyce

Solutions for Chapter 5.4: Euler Equations; Regular Singular Points

Solutions for Chapter 5.4
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Textbook: Elementary Differential Equations and Boundary Value Problems
Edition: 10
Author: William E. Boyce
ISBN: 9780470458310

This expansive textbook survival guide covers the following chapters and their solutions. Chapter 5.4: Euler Equations; Regular Singular Points includes 49 full step-by-step solutions. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470458310. Since 49 problems in chapter 5.4: Euler Equations; Regular Singular Points have been answered, more than 16837 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 10.

Key Math Terms and definitions covered in this textbook
  • Adjacency matrix of a graph.

    Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

  • Back substitution.

    Upper triangular systems are solved in reverse order Xn to Xl.

  • Cofactor Cij.

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

  • Column space C (A) =

    space of all combinations of the columns of A.

  • Diagonal matrix D.

    dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.

  • Dimension of vector space

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

  • Free columns of A.

    Columns without pivots; these are combinations of earlier columns.

  • Gram-Schmidt orthogonalization A = QR.

    Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

  • 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.

  • 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).

  • Nullspace matrix N.

    The columns of N are the n - r special solutions to As = O.

  • Plane (or hyperplane) in Rn.

    Vectors x with aT x = O. Plane is perpendicular to a =1= O.

  • Projection matrix P onto subspace S.

    Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.

  • Rotation matrix

    R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().

  • Trace of A

    = sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

  • 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·

  • Tridiagonal matrix T: tij = 0 if Ii - j I > 1.

    T- 1 has rank 1 above and below diagonal.

  • Vector space V.

    Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

  • Wavelets Wjk(t).

    Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).

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