 5.3.1: In each of 1 through 4, determine (x0), (x0), and (4)(x0) for the g...
 5.3.2: In each of 1 through 4, determine (x0), (x0), and (4)(x0) for the g...
 5.3.3: In each of 1 through 4, determine (x0), (x0), and (4)(x0) for the g...
 5.3.4: In each of 1 through 4, determine (x0), (x0), and (4)(x0) for the g...
 5.3.5: In each of 5 through 8, determine a lower bound for the radius of c...
 5.3.6: In each of 5 through 8, determine a lower bound for the radius of c...
 5.3.7: In each of 5 through 8, determine a lower bound for the radius of c...
 5.3.8: In each of 5 through 8, determine a lower bound for the radius of c...
 5.3.9: Determine a lower bound for the radius of convergence of series sol...
 5.3.10: The Chebyshev Equation. The Chebyshev7 differential equation is(1 x...
 5.3.11: For each of the differential equations in 11 through 14, find the f...
 5.3.12: For each of the differential equations in 11 through 14, find the f...
 5.3.13: For each of the differential equations in 11 through 14, find the f...
 5.3.14: For each of the differential equations in 11 through 14, find the f...
 5.3.15: Let x and x2 be solutions of a differential equation P(x)y + Q(x)y ...
 5.3.16: First Order Equations. The series methods discussed in this section...
 5.3.17: First Order Equations. The series methods discussed in this section...
 5.3.18: First Order Equations. The series methods discussed in this section...
 5.3.19: First Order Equations. The series methods discussed in this section...
 5.3.20: First Order Equations. The series methods discussed in this section...
 5.3.21: First Order Equations. The series methods discussed in this section...
 5.3.22: The Legendre Equation. 22 through 29 deal with the Legendre8 equati...
 5.3.23: The Legendre Equation. 22 through 29 deal with the Legendre8 equati...
 5.3.24: The Legendre Equation. 22 through 29 deal with the Legendre8 equati...
 5.3.25: The Legendre Equation. 22 through 29 deal with the Legendre8 equati...
 5.3.26: The Legendre Equation. 22 through 29 deal with the Legendre8 equati...
 5.3.27: The Legendre Equation. 22 through 29 deal with the Legendre8 equati...
 5.3.28: The Legendre Equation. 22 through 29 deal with the Legendre8 equati...
 5.3.29: The Legendre Equation. 22 through 29 deal with the Legendre8 equati...
Solutions for Chapter 5.3: Series Solutions Near an Ordinary Point, Part II
Full solutions for Elementary Differential Equations  10th Edition
ISBN: 9780470458327
Solutions for Chapter 5.3: Series Solutions Near an Ordinary Point, Part II
Get Full SolutionsChapter 5.3: Series Solutions Near an Ordinary Point, Part II includes 29 full stepbystep solutions. 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 11633 students have viewed full stepbystep solutions from this chapter. Elementary Differential Equations was written by and is associated to the ISBN: 9780470458327. This textbook survival guide was created for the textbook: Elementary Differential Equations, edition: 10.

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.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

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.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. 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 combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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

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 •

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Solvable system Ax = b.
The right side b is in the column space of A.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

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