 5.6.1: In each of 1 through 12:(a) Find all the regular singular points of...
 5.6.2: In each of 1 through 12:(a) Find all the regular singular points of...
 5.6.3: In each of 1 through 12:(a) Find all the regular singular points of...
 5.6.4: In each of 1 through 12:(a) Find all the regular singular points of...
 5.6.5: In each of 1 through 12:(a) Find all the regular singular points of...
 5.6.6: In each of 1 through 12:(a) Find all the regular singular points of...
 5.6.7: In each of 1 through 12:(a) Find all the regular singular points of...
 5.6.8: In each of 1 through 12:(a) Find all the regular singular points of...
 5.6.9: In each of 1 through 12:(a) Find all the regular singular points of...
 5.6.10: In each of 1 through 12:(a) Find all the regular singular points of...
 5.6.11: In each of 1 through 12:(a) Find all the regular singular points of...
 5.6.12: In each of 1 through 12:(a) Find all the regular singular points of...
 5.6.13: In each of 13 through 17:(a) Show that x = 0 is a regular singular ...
 5.6.14: In each of 13 through 17:(a) Show that x = 0 is a regular singular ...
 5.6.15: In each of 13 through 17:(a) Show that x = 0 is a regular singular ...
 5.6.16: In each of 13 through 17:(a) Show that x = 0 is a regular singular ...
 5.6.17: In each of 13 through 17:(a) Show that x = 0 is a regular singular ...
 5.6.18: (a) Show that(ln x)y+ 12 y+ y = 0has a regular singular point at x ...
 5.6.19: In several problems in mathematical physics, it is necessary to stu...
 5.6.20: Consider the differential equationx3y+ xy+ y = 0,where and are real...
 5.6.21: Consider the differential equationy+ xs y+ xty = 0, (i)where = 0 an...
Solutions for Chapter 5.6: Series Solutions Near a Regular Singular Point, Part II
Full solutions for Elementary Differential Equations and Boundary Value Problems  10th Edition
ISBN: 9780470458310
Solutions for Chapter 5.6: Series Solutions Near a Regular Singular Point, Part II
Get Full SolutionsElementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470458310. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 10. Chapter 5.6: Series Solutions Near a Regular Singular Point, Part II includes 21 full stepbystep solutions. Since 21 problems in chapter 5.6: Series Solutions Near a Regular Singular Point, Part II have been answered, more than 16347 students have viewed full stepbystep solutions from this chapter.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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 .

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.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Outer product uv T
= column times row = rank one matrix.

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.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

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

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.