 6.4.6.1.108: In 16 use (1) to find the general solution of the given differentia...
 6.4.6.1.109: In 16 use (1) to find the general solution of the given differentia...
 6.4.6.1.110: In 16 use (1) to find the general solution of the given differentia...
 6.4.6.1.111: In 16 use (1) to find the general solution of the given differentia...
 6.4.6.1.112: In 16 use (1) to find the general solution of the given differentia...
 6.4.6.1.113: In 16 use (1) to find the general solution of the given differentia...
 6.4.6.1.114: In 710 use (12) to find the general solution of the given different...
 6.4.6.1.115: In 710 use (12) to find the general solution of the given different...
 6.4.6.1.116: In 710 use (12) to find the general solution of the given different...
 6.4.6.1.117: In 710 use (12) to find the general solution of the given different...
 6.4.6.1.118: In 11 and 12 use the indicated change of variable to find the gener...
 6.4.6.1.119: In 11 and 12 use the indicated change of variable to find the gener...
 6.4.6.1.120: In 1320 use (18) to find the general solution of the given differen...
 6.4.6.1.121: In 1320 use (18) to find the general solution of the given differen...
 6.4.6.1.122: In 1320 use (18) to find the general solution of the given differen...
 6.4.6.1.123: In 1320 use (18) to find the general solution of the given differen...
 6.4.6.1.124: In 1320 use (18) to find the general solution of the given differen...
 6.4.6.1.125: In 1320 use (18) to find the general solution of the given differen...
 6.4.6.1.126: In 1320 use (18) to find the general solution of the given differen...
 6.4.6.1.127: In 1320 use (18) to find the general solution of the given differen...
 6.4.6.1.128: Use the series in (7) to verify that I(x) i J(ix) is a real function.
 6.4.6.1.129: Assume that b in equation (18) can be pure imaginary, that is, b i,...
 6.4.6.1.130: In 2326 first use (18) to express the general solution of the given...
 6.4.6.1.131: In 2326 first use (18) to express the general solution of the given...
 6.4.6.1.132: In 2326 first use (18) to express the general solution of the given...
 6.4.6.1.133: In 2326 first use (18) to express the general solution of the given...
 6.4.6.1.134: (a) Proceed as in Example 5 to show that xJ (x) J(x) xJ1(x). [Hint:...
 6.4.6.1.135: Use the formula obtained in Example 5 along with part (a) of to der...
 6.4.6.1.136: In 29 and 30 use (20) or (21) to obtain the given result. x 0 rJ0(r...
 6.4.6.1.137: In 29 and 30 use (20) or (21) to obtain the given result.J0 (x) J 1...
 6.4.6.1.138: Proceed as on page 264 to derive the elementary form of J1/2(x) giv...
 6.4.6.1.139: Use the recurrence relation in along with (23) and (24) to express ...
 6.4.6.1.140: Use the change of variables to show that the differential equation ...
 6.4.6.1.141: Show that is a solution of Airys differential equation y 2xy 0, x 0...
 6.4.6.1.142: (a) Use the result of to express the general solution of Airys diff...
 6.4.6.1.143: Use the Table 6.4.1 to find the first three positive eigenvalues an...
 6.4.6.1.144: (a) Use (18) to show that the general solution of the differential ...
 6.4.6.1.145: Use a CAS to graph J3/2(x), J3/2(x), J5/2(x), and J5/2(x).
 6.4.6.1.146: (a) Use the general solution given in Example 4 to solve the IVP Al...
 6.4.6.1.147: (a) Use the general solution obtained in to solve the IVP Use a CAS...
 6.4.6.1.148: Column Bending Under Its Own Weight A uniform thin column of length...
 6.4.6.1.149: Buckling of a Thin Vertical Column In Example 4 of Section 5.2 we s...
 6.4.6.1.150: Pendulum of Varying Length For the simple pendulum described on pag...
 6.4.6.1.151: (a) Use the explicit solutions y1(x) and y2(x) of Legendres equatio...
 6.4.6.1.152: Use the recurrence relation (32) and P0(x) 1, P1(x) x, to generate ...
 6.4.6.1.153: Show that the differential equation can be transformed into Legendr...
 6.4.6.1.154: Find the first three positive values of for which the problem has n...
 6.4.6.1.155: For purposes of this problem ignore the list of Legendre polynomial...
 6.4.6.1.156: Use a CAS to graph P1(x), P2(x), . . . , P7 (x) on the interval [1,...
 6.4.6.1.157: Use a rootfindin application to fin the zeros of P1(x), P2(x), . ....
 6.4.6.1.158: The differential equation y 2xy 2ay 0 is known as Hermites equation...
 6.4.6.1.159: (a) When is a nonnegative integer, Hermites differential equation a...
 6.4.6.1.160: The differential equation , where is a parameter, is known as Cheby...
 6.4.6.1.161: If n is an integer, use the substitution to show that the general s...
Solutions for Chapter 6.4: Series Solutions of Linear Equations
Full solutions for Differential Equations with BoundaryValue Problems,  8th Edition
ISBN: 9781111827069
Solutions for Chapter 6.4: Series Solutions of Linear Equations
Get Full SolutionsSince 54 problems in chapter 6.4: Series Solutions of Linear Equations have been answered, more than 20480 students have viewed full stepbystep solutions from this chapter. Differential Equations with BoundaryValue Problems, was written by and is associated to the ISBN: 9781111827069. Chapter 6.4: Series Solutions of Linear Equations includes 54 full stepbystep solutions. This textbook survival guide was created for the textbook: Differential Equations with BoundaryValue Problems,, edition: 8. 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.

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

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).

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

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Iterative method.
A sequence of steps intended to approach the desired solution.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

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.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

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.

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.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

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

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

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

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).