 2.7.1: Most of the problems in this section call for fairly extensive nume...
 2.7.2: Most of the problems in this section call for fairly extensive nume...
 2.7.3: Most of the problems in this section call for fairly extensive nume...
 2.7.4: Most 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 diffe...
 2.7.6: In each of 5 through 10, draw a direction field for the given diffe...
 2.7.7: In each of 5 through 10, draw a direction field for the given diffe...
 2.7.8: In each of 5 through 10, draw a direction field for the given diffe...
 2.7.9: In each of 5 through 10, draw a direction field for the given diffe...
 2.7.10: In each of 5 through 10, draw a direction field for the given diffe...
 2.7.11: In each of 11 through 14, use Eulers method to find approximate val...
 2.7.12: In each of 11 through 14, use Eulers method to find approximate val...
 2.7.13: In each of 11 through 14, use Eulers method to find approximate val...
 2.7.14: In each of 11 through 14, use Eulers method to find approximate val...
 2.7.15: Consider the initial value problemy= 3t2/(3y2 4), y(1) = 0.(a) Use ...
 2.7.16: Consider the initial value problemy= t2 + y2, y(0) = 1.Use Eulers m...
 2.7.17: Consider the initial value problemy= (y2 + 2ty)/(3 + t2), y(1) = 2....
 2.7.18: Consider the initial value problemy= ty + 0.1y3, y(0) = ,where is a...
 2.7.19: Consider the initial value problemy= y2 t2, y(0) = ,where is a give...
 2.7.20: Convergence of Eulers Method. It can be shown that under suitable c...
 2.7.21: In each of 21 through 23, use the technique discussed in to show th...
 2.7.22: In each of 21 through 23, use the technique discussed in to show th...
 2.7.23: In each of 21 through 23, use the technique discussed in to show th...
Solutions for Chapter 2.7: Numerical Approximations: Eulers Method
Full solutions for Elementary Differential Equations and Boundary Value Problems  10th Edition
ISBN: 9780470458310
Solutions for Chapter 2.7: Numerical Approximations: Eulers Method
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 10. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470458310. Chapter 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 16365 students have viewed full stepbystep solutions from this chapter.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

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.

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

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Iterative method.
A sequence of steps intended to approach the desired solution.

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.

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.

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.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.