 5.3.1: In each of 1 through 4, determine (x0), (x0), and (4) (x0) for the ...
 5.3.2: In each of 1 through 4, determine (x0), (x0), and (4) (x0) for the ...
 5.3.3: In each of 1 through 4, determine (x0), (x0), and (4) (x0) for the ...
 5.3.4: In each of 1 through 4, determine (x0), (x0), and (4) (x0) for the ...
 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 and Boundary Value Problems  10th Edition
ISBN: 9780470458310
Solutions for Chapter 5.3: Series Solutions Near an Ordinary Point, Part II
Get Full SolutionsThis 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 stepbystep 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 stepbystep solutions. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470458310.

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 IIA1 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.

lAII = 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 AI are BT AT and (AT)I.

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

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