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Solutions for Chapter 1.9: Exact equations, and why we cannot solve very many differential equations

Differential Equations and Their Applications: An Introduction to Applied Mathematics | 3rd Edition | ISBN: 9780387908069 | Authors: M. Braun

Full solutions for Differential Equations and Their Applications: An Introduction to Applied Mathematics | 3rd Edition

ISBN: 9780387908069

Differential Equations and Their Applications: An Introduction to Applied Mathematics | 3rd Edition | ISBN: 9780387908069 | Authors: M. Braun

Solutions for Chapter 1.9: Exact equations, and why we cannot solve very many differential equations

Solutions for Chapter 1.9
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Textbook: Differential Equations and Their Applications: An Introduction to Applied Mathematics
Edition: 3
Author: M. Braun
ISBN: 9780387908069

Since 20 problems in chapter 1.9: Exact equations, and why we cannot solve very many differential equations have been answered, more than 5912 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Differential Equations and Their Applications: An Introduction to Applied Mathematics, edition: 3. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 1.9: Exact equations, and why we cannot solve very many differential equations includes 20 full step-by-step solutions. Differential Equations and Their Applications: An Introduction to Applied Mathematics was written by and is associated to the ISBN: 9780387908069.

Key Math Terms and definitions covered in this textbook
  • Affine transformation

    Tv = Av + Vo = linear transformation plus shift.

  • Associative Law (AB)C = A(BC).

    Parentheses can be removed to leave ABC.

  • Change of basis matrix M.

    The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

  • 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).

  • Column space C (A) =

    space of all combinations of the columns of A.

  • Conjugate Gradient Method.

    A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.

  • Diagonal matrix D.

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

  • 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

  • Free columns of A.

    Columns without pivots; these are combinations of earlier columns.

  • Free variable Xi.

    Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

  • Graph G.

    Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.

  • 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.

  • Normal matrix.

    If N NT = NT N, then N has orthonormal (complex) eigenvectors.

  • Nullspace matrix N.

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

  • Orthonormal vectors q 1 , ... , q n·

    Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

  • 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 ].

  • Trace of A

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

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

    T- 1 has rank 1 above and below diagonal.

  • Vector v in Rn.

    Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

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