 11.3.1: Use Equations (11.3.3) and (11.3.4) to determine polynomial solutio...
 11.3.2: Starting with P0(x) = 1 and P1(x) = x, use the recurrence relation ...
 11.3.3: Use Rodrigues formula to determine the Legendre polynomial of degre...
 11.3.4: Determine all values of the constants a0, a1, a2, and a3 such that ...
 11.3.5: Express p(x) = 2x3 + x2 + 5 as a linear combination of Legendre pol...
 11.3.6: Let Q(x) be a polynomial of degree less than N. Prove that 1 1 Q(x)...
 11.3.7: Show that d2Y d2 + cot dY d + ( + 1)Y = 0, 0 < < , is transformed i...
 11.3.8: Determine two linearly independent series solutions to Hermites equ...
 11.3.9: Show that if = N, a positive integer, then Equation (11.3.13) has a...
 11.3.10: When suitably normalized, the polynomial solutions to Equation (11....
 11.3.11: Use some form of technology to determine the coef ficients in the ...
 11.3.12: For 1213, use some form of technology to determine the first four t...
 11.3.13: For 1213, use some form of technology to determine the first four t...
Solutions for Chapter 11.3: The Legendre Equation
Full solutions for Differential Equations  4th Edition
ISBN: 9780321964670
Solutions for Chapter 11.3: The Legendre Equation
Get Full SolutionsSince 13 problems in chapter 11.3: The Legendre Equation have been answered, more than 20058 students have viewed full stepbystep solutions from this chapter. Differential Equations was written by and is associated to the ISBN: 9780321964670. 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. Chapter 11.3: The Legendre Equation includes 13 full stepbystep solutions.

Column space C (A) =
space of all combinations of the columns of A.

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.

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.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

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

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

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.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

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

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

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

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

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

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.

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