 5.5.1: In each of 1 through 10:(a) Show that the given differential equati...
 5.5.2: In each of 1 through 10:(a) Show that the given differential equati...
 5.5.3: In each of 1 through 10:(a) Show that the given differential equati...
 5.5.4: In each of 1 through 10:(a) Show that the given differential equati...
 5.5.5: In each of 1 through 10:(a) Show that the given differential equati...
 5.5.6: In each of 1 through 10:(a) Show that the given differential equati...
 5.5.7: In each of 1 through 10:(a) Show that the given differential equati...
 5.5.8: In each of 1 through 10:(a) Show that the given differential equati...
 5.5.9: In each of 1 through 10:(a) Show that the given differential equati...
 5.5.10: In each of 1 through 10:(a) Show that the given differential equati...
 5.5.11: The Legendre equation of order is(1 x2)y 2xy + ( + 1)y = 0.The solu...
 5.5.12: The Chebyshev equation is(1 x2)y xy + 2y = 0,where is a constant; s...
 5.5.13: The Laguerre13 differential equation isxy + (1 x)y + y = 0.(a) Show...
 5.5.14: The Bessel equation of order zero isx2y + xy + x2y = 0.(a) Show tha...
 5.5.15: Referring to 14, use the method of reduction of order to show that ...
 5.5.16: The Bessel equation of order one isx2y + xy + (x2 1)y = 0.(a) Show ...
Solutions for Chapter 5.5: Series Solutions Near a Regular Singular Point, Part I
Full solutions for Elementary Differential Equations  10th Edition
ISBN: 9780470458327
Solutions for Chapter 5.5: Series Solutions Near a Regular Singular Point, Part I
Get Full SolutionsElementary Differential Equations was written by and is associated to the ISBN: 9780470458327. Chapter 5.5: Series Solutions Near a Regular Singular Point, Part I includes 16 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Differential Equations, edition: 10. Since 16 problems in chapter 5.5: Series Solutions Near a Regular Singular Point, Part I have been answered, more than 12175 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

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

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

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

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

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

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

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

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

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