 11.7.1: For 113 determine whether x = 0 is an ordinary point or a regular s...
 11.7.2: For 113 determine whether x = 0 is an ordinary point or a regular s...
 11.7.3: For 113 determine whether x = 0 is an ordinary point or a regular s...
 11.7.4: For 113 determine whether x = 0 is an ordinary point or a regular s...
 11.7.5: For 113 determine whether x = 0 is an ordinary point or a regular s...
 11.7.6: For 113 determine whether x = 0 is an ordinary point or a regular s...
 11.7.7: For 113 determine whether x = 0 is an ordinary point or a regular s...
 11.7.8: For 113 determine whether x = 0 is an ordinary point or a regular s...
 11.7.9: For 113 determine whether x = 0 is an ordinary point or a regular s...
 11.7.10: For 113 determine whether x = 0 is an ordinary point or a regular s...
 11.7.11: For 113 determine whether x = 0 is an ordinary point or a regular s...
 11.7.12: For 113 determine whether x = 0 is an ordinary point or a regular s...
 11.7.13: For 113 determine whether x = 0 is an ordinary point or a regular s...
 11.7.14: Consider the hypergeometric equation x(1 x)y + [c (a + b + 1)x]y ab...
 11.7.15: Consider the differential equation (x2 1)y + [1 (a + b)]x y + aby =...
 11.7.16: Consider the Chebyshev equation (1 x2)y x y + a2 y = 0, (11.7.10) w...
 11.7.17: Consider the differential equation x2 y + x(1 + 2N x)y + N2 y = 0, ...
 11.7.18: Consider the general perturbed CauchyEuler equation x2 y + x[1 (a ...
 11.7.19: Consider the differential equation x2 y + x(1 + bx)y + [b(1 N)x N2]...
 11.7.20: Show that the change of variables y = x1/2u transforms the differen...
Solutions for Chapter 11.7: Series Solutions to Linear Differential Equations
Full solutions for Differential Equations  4th Edition
ISBN: 9780321964670
Solutions for Chapter 11.7: Series Solutions to Linear Differential Equations
Get Full SolutionsChapter 11.7: Series Solutions to Linear Differential Equations includes 20 full stepbystep solutions. Differential Equations was written by and is associated to the ISBN: 9780321964670. Since 20 problems in chapter 11.7: Series Solutions to Linear Differential Equations have been answered, more than 20006 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Differential Equations, edition: 4. This expansive textbook survival guide covers the following chapters and their solutions.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

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

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

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

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.

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

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).

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

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

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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.

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

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

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

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

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