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Solutions for Chapter 5.3: Series Solutions Near an Ordinary Point, Part II

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 5.3: Series Solutions Near an Ordinary Point, Part II

Solutions for Chapter 5.3
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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. Since 29 problems in chapter 5.3: Series Solutions Near an Ordinary Point, Part II have been answered, more than 16358 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. Chapter 5.3: Series Solutions Near an Ordinary Point, Part II includes 29 full step-by-step solutions. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470458310.

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).

  • Basis for V.

    Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

  • 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.

  • 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).

  • Ellipse (or ellipsoid) x T Ax = 1.

    A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad

  • 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.

  • lA-II = l/lAI and IATI = IAI.

    The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.

  • 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.

  • Markov matrix M.

    All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

  • Minimal polynomial of A.

    The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).

  • Nullspace N (A)

    = All solutions to Ax = O. Dimension n - r = (# columns) - rank.

  • Projection p = a(aTblaTa) onto the line through a.

    P = aaT laTa has rank l.

  • Rank one matrix A = uvT f=. O.

    Column and row spaces = lines cu and cv.

  • Rank r (A)

    = number of pivots = dimension of column space = dimension of row space.

  • Right inverse A+.

    If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.

  • Schur complement S, D - C A -} B.

    Appears in block elimination on [~ g ].

  • Singular matrix A.

    A square matrix that has no inverse: det(A) = o.

  • 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.

  • Unitary matrix UH = U T = U-I.

    Orthonormal columns (complex analog of Q).

  • Wavelets Wjk(t).

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

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