- 5.7.1: In each of 1 through 4, show that the given differential equation h...
- 5.7.2: In each of 1 through 4, show that the given differential equation h...
- 5.7.3: In each of 1 through 4, show that the given differential equation h...
- 5.7.4: In each of 1 through 4, show that the given differential equation h...
- 5.7.5: Find two solutions (not multiples of each other) of the Bessel equa...
- 5.7.6: Show that the Bessel equation of order one-halfx2y + xy + x2 14y = ...
- 5.7.7: Show directly that the series for J0(x), Eq. (7), converges absolut...
- 5.7.8: Show directly that the series for J1(x), Eq. (27), converges absolu...
- 5.7.9: Consider the Bessel equation of order x2y + xy + (x2 2)y = 0, x > 0...
- 5.7.10: In this section we showed that one solution of Bessels equation of ...
- 5.7.11: Find a second solution of Bessels equation of order one by computin...
- 5.7.12: By a suitable change of variables it is sometimes possible to trans...
- 5.7.13: Using the result of 12, show that the general solution of the Airy ...
- 5.7.14: It can be shown that J0 has infinitely many zeros for x > 0. In par...
Solutions for Chapter 5.7: Bessels Equation
Full solutions for Elementary Differential Equations | 10th Edition
Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).
Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.
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.
peA) = det(A - AI) has peA) = zero matrix.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
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.
Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.
Reflection matrix (Householder) Q = I -2uuT.
Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.