 9.3.9.1.43: Find the analytic solution of the initialvalue problem in Example ...
 9.3.9.1.44: Write a computer program to implement the AdamsBashforthMoulton me...
 9.3.9.1.45: In 3 and 4 use the AdamsBashforthMoulton method to approximate y(...
 9.3.9.1.46: In 3 and 4 use the AdamsBashforthMoulton method to approximate y(...
 9.3.9.1.47: In 58 use the AdamsBashforthMoulton method to approximate y(1.0),...
 9.3.9.1.48: In 58 use the AdamsBashforthMoulton method to approximate y(1.0),...
 9.3.9.1.49: In 58 use the AdamsBashforthMoulton method to approximate y(1.0),...
 9.3.9.1.50: In 58 use the AdamsBashforthMoulton method to approximate y(1.0),...
Solutions for Chapter 9.3: Numerical Solutions of Ordinary Differential Equations
Full solutions for Differential Equations with BoundaryValue Problems,  8th Edition
ISBN: 9781111827069
Solutions for Chapter 9.3: Numerical Solutions of Ordinary Differential Equations
Get Full SolutionsSince 8 problems in chapter 9.3: Numerical Solutions of Ordinary Differential Equations have been answered, more than 23170 students have viewed full stepbystep solutions from this chapter. Differential Equations with BoundaryValue Problems, was written by and is associated to the ISBN: 9781111827069. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 9.3: Numerical Solutions of Ordinary Differential Equations includes 8 full stepbystep solutions. This textbook survival guide was created for the textbook: Differential Equations with BoundaryValue Problems,, edition: 8.

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).

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

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.

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.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

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