 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 co...
 5.3.6: In each of 5 through 8 determine a lower bound for the radius of co...
 5.3.7: In each of 5 through 8 determine a lower bound for the radius of co...
 5.3.8: In each of 5 through 8 determine a lower bound for the radius of co...
 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 ...
 5.3.11: For each of the differential equations in 11 through 14 find the fi...
 5.3.12: For each of the differential equations in 11 through 14 find the fi...
 5.3.13: For each of the differential equations in 11 through 14 find the fi...
 5.3.14: For each of the differential equations in 11 through 14 find the fi...
 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: Show that, if is zero or a positive even integer 2n, the series sol...
 5.3.24: The Legendre polynomial Pn(x) is defined as the polynomial solution...
 5.3.25: It can be shown that the general formula for Pn(x) is Pn(x) = 1 2n ...
 5.3.26: The Legendre polynomials play an important role in mathematical phy...
 5.3.27: Show that for n = 0, 1, 2, 3, the corresponding Legendre polynomial...
 5.3.28: Show that the Legendre equation can also be written as [(1 x2 )y ] ...
 5.3.29: 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  9th Edition
ISBN: 9780470383346
Solutions for Chapter 5.3: Series Solutions Near an Ordinary Point, Part II
Get Full SolutionsThis textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 9. Chapter 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 12713 students have viewed full stepbystep solutions from this chapter. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470383346.

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.

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].

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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).

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).

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.

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

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

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