 9.1.9.1.1: In 110 use the improved Eulers method to obtain a fourdecimal appr...
 9.1.9.1.2: In 110 use the improved Eulers method to obtain a fourdecimal appr...
 9.1.9.1.3: In 110 use the improved Eulers method to obtain a fourdecimal appr...
 9.1.9.1.4: In 110 use the improved Eulers method to obtain a fourdecimal appr...
 9.1.9.1.5: In 110 use the improved Eulers method to obtain a fourdecimal appr...
 9.1.9.1.6: In 110 use the improved Eulers method to obtain a fourdecimal appr...
 9.1.9.1.7: In 110 use the improved Eulers method to obtain a fourdecimal appr...
 9.1.9.1.8: In 110 use the improved Eulers method to obtain a fourdecimal appr...
 9.1.9.1.9: In 110 use the improved Eulers method to obtain a fourdecimal appr...
 9.1.9.1.10: In 110 use the improved Eulers method to obtain a fourdecimal appr...
 9.1.9.1.11: Consider the initialvalue problem y (x y 1)2 , y(0) 2. Use the imp...
 9.1.9.1.12: Although it might not be obvious from the differential equation, it...
 9.1.9.1.13: Consider the initialvalue problem y 2y, y(0) 1. The analytic solut...
 9.1.9.1.14: Repeat using the improved Eulers method. Its global truncation erro...
 9.1.9.1.15: Repeat using the initialvalue problem y x 2y, y(0) 1. The analytic...
 9.1.9.1.16: Repeat using the improved Eulers method. Its global truncation erro...
 9.1.9.1.17: Consider the initialvalue problem y 2x 3y 1, y(1) 5. The analytic ...
 9.1.9.1.18: Repeat using the improved Eulers method, which has a global truncat...
 9.1.9.1.19: Repeat for the initialvalue problem y ey , y(0) 0. The analytic so...
 9.1.9.1.20: Repeat using the improved Eulers method, which has global truncatio...
 9.1.9.1.21: Answer the question Why not? that follows the three sentences after...
Solutions for Chapter 9.1: Numerical Solutions of Ordinary Differential Equations
Full solutions for Differential Equations with BoundaryValue Problems,  8th Edition
ISBN: 9781111827069
Solutions for Chapter 9.1: Numerical Solutions of Ordinary Differential Equations
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 9.1: Numerical Solutions of Ordinary Differential Equations includes 21 full stepbystep solutions. Differential Equations with BoundaryValue Problems, was written by and is associated to the ISBN: 9781111827069. This textbook survival guide was created for the textbook: Differential Equations with BoundaryValue Problems,, edition: 8. Since 21 problems in chapter 9.1: Numerical Solutions of Ordinary Differential Equations have been answered, more than 20452 students have viewed full stepbystep solutions from this chapter.

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.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

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).

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.

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.

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

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

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).

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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

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.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

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.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

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

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·