 5.3.1: In each of 1 through 3, determine ( x0), ( x0), and (4) ( x0) for t...
 5.3.2: In each of 1 through 3, determine ( x0), ( x0), and (4) ( x0) for t...
 5.3.3: In each of 1 through 3, determine ( x0), ( x0), and (4) ( x0) for t...
 5.3.4: In each of 4 through 6, determine a lower bound for the radius of c...
 5.3.5: In each of 4 through 6, determine a lower bound for the radius of c...
 5.3.6: In each of 4 through 6, determine a lower bound for the radius of c...
 5.3.7: Determine a lower bound for the radius of convergence of series sol...
 5.3.8: The Chebyshev Equation. The Chebyshev7 differential equation is (1 ...
 5.3.9: For each of the differential equations in 9 through 11, find the fi...
 5.3.10: For each of the differential equations in 9 through 11, find the fi...
 5.3.11: For each of the differential equations in 9 through 11, find the fi...
 5.3.12: Let y = x and y = x2 be solutions of a differential equation P( x) ...
 5.3.13: In each of 13 through 16, solve the given differential equation by ...
 5.3.14: In each of 13 through 16, solve the given differential equation by ...
 5.3.15: In each of 13 through 16, solve the given differential equation by ...
 5.3.16: In each of 13 through 16, solve the given differential equation by ...
 5.3.17: Show that two solutions of the Legendre equation for x < 1 are y1...
 5.3.18: Show that if is zero or a positive even integer 2n, the series solu...
 5.3.19: The Legendre polynomial Pn( x) is defined as the polynomial solutio...
 5.3.20: The Legendre polynomials play an important role in mathematical phy...
 5.3.21: Show that for n = 0, 1, 2, 3, the corresponding Legendre polynomial...
 5.3.22: Show that the Legendre equation can also be written as (1 x2) y__ _...
 5.3.23: Given a polynomial f of degree n, it is possible to express f as a ...
Solutions for Chapter 5.3: Series Solutions Near an Ordinary Point, Part II
Full solutions for Elementary Differential Equations and Boundary Value Problems  11th Edition
ISBN: 9781119256007
Solutions for Chapter 5.3: Series Solutions Near an Ordinary Point, Part II
Get Full SolutionsSince 23 problems in chapter 5.3: Series Solutions Near an Ordinary Point, Part II have been answered, more than 12334 students have viewed full stepbystep solutions from this chapter. Chapter 5.3: Series Solutions Near an Ordinary Point, Part II includes 23 full stepbystep solutions. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9781119256007. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 11. This expansive textbook survival guide covers the following chapters and their solutions.

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.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

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.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

GramSchmidt 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.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

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.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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).