 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 onehalfx2y + 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
ISBN: 9780470458327
Solutions for Chapter 5.7: Bessels Equation
Get Full SolutionsElementary Differential Equations was written by and is associated to the ISBN: 9780470458327. Since 14 problems in chapter 5.7: Bessels Equation have been answered, more than 11238 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Elementary Differential Equations, edition: 10. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 5.7: Bessels Equation includes 14 full stepbystep solutions.

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.

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.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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

Distributive Law
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 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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 = Q1 = 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. AI is also symmetric.