 2.7.1: Many of the problems in this section call for fairly extensive nume...
 2.7.2: Many of the problems in this section call for fairly extensive nume...
 2.7.3: Many of the problems in this section call for fairly extensive nume...
 2.7.4: Many of the problems in this section call for fairly extensive nume...
 2.7.5: In each of 5 through 10 draw a direction field for the given differ...
 2.7.6: In each of 5 through 10 draw a direction field for the given differ...
 2.7.7: In each of 5 through 10 draw a direction field for the given differ...
 2.7.8: In each of 5 through 10 draw a direction field for the given differ...
 2.7.9: In each of 5 through 10 draw a direction field for the given differ...
 2.7.10: In each of 5 through 10 draw a direction field for the given differ...
 2.7.11: In each of 11 through 14 use Eulers method to find approximate valu...
 2.7.12: In each of 11 through 14 use Eulers method to find approximate valu...
 2.7.13: In each of 11 through 14 use Eulers method to find approximate valu...
 2.7.14: In each of 11 through 14 use Eulers method to find approximate valu...
 2.7.15: Consider the initial value problem y = 3t 2 /(3y2 4), y(1) = 0. (a)...
 2.7.16: Consider the initial value problem y = t 2 + y2 , y(0) = 1. Use Eul...
 2.7.17: Consider the initial value problem y = (y2 + 2ty)/(3 + t 2 ), y(1) ...
 2.7.18: Consider the initial value problem y = ty + 0.1y3 , y(0) = , where ...
 2.7.19: Consider the initial value problem y = y2 t 2 , y(0) = , where is a...
 2.7.20: Convergence of Eulers Method. It can be shown that, under suitable ...
 2.7.21: In each of 21 through 23 use the technique discussed in to show tha...
 2.7.22: In each of 21 through 23 use the technique discussed in to show tha...
 2.7.23: In each of 21 through 23 use the technique discussed in to show tha...
Solutions for Chapter 2.7: Numerical Approximations: Eulers Method
Full solutions for Elementary Differential Equations and Boundary Value Problems  9th Edition
ISBN: 9780470383346
Solutions for Chapter 2.7: Numerical Approximations: Eulers Method
Get Full SolutionsChapter 2.7: Numerical Approximations: Eulers Method includes 23 full stepbystep solutions. Since 23 problems in chapter 2.7: Numerical Approximations: Eulers Method have been answered, more than 9660 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470383346. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 9.

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

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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).

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.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

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.

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

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.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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

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

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

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

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.