 2.4.1: In 1 through 10, an initial value problem and its exact solution y....
 2.4.2: In 1 through 10, an initial value problem and its exact solution y....
 2.4.3: In 1 through 10, an initial value problem and its exact solution y....
 2.4.4: In 1 through 10, an initial value problem and its exact solution y....
 2.4.5: In 1 through 10, an initial value problem and its exact solution y....
 2.4.6: In 1 through 10, an initial value problem and its exact solution y....
 2.4.7: In 1 through 10, an initial value problem and its exact solution y....
 2.4.8: In 1 through 10, an initial value problem and its exact solution y....
 2.4.9: In 1 through 10, an initial value problem and its exact solution y....
 2.4.10: In 1 through 10, an initial value problem and its exact solution y....
 2.4.11: A programmable calculator or a computer will be useful for 11 throu...
 2.4.12: A programmable calculator or a computer will be useful for 11 throu...
 2.4.13: A programmable calculator or a computer will be useful for 11 throu...
 2.4.14: A programmable calculator or a computer will be useful for 11 throu...
 2.4.15: A programmable calculator or a computer will be useful for 11 throu...
 2.4.16: A programmable calculator or a computer will be useful for 11 throu...
 2.4.17: A computer with a printer is required for 17 through 24. In these i...
 2.4.18: A computer with a printer is required for 17 through 24. In these i...
 2.4.19: A computer with a printer is required for 17 through 24. In these i...
 2.4.20: A computer with a printer is required for 17 through 24. In these i...
 2.4.21: A computer with a printer is required for 17 through 24. In these i...
 2.4.22: A computer with a printer is required for 17 through 24. In these i...
 2.4.23: A computer with a printer is required for 17 through 24. In these i...
 2.4.24: A computer with a printer is required for 17 through 24. In these i...
 2.4.25: You bail out of the helicopter of Example 2 and immediately pull th...
 2.4.26: Suppose the deer population P .t / in a small forest initially numb...
 2.4.27: Use Eulers method with a computer system to find the desired soluti...
 2.4.28: Use Eulers method with a computer system to find the desired soluti...
 2.4.29: . Consider the initial value problem 7x dy dx C y D 0; y.1/ D 1: (a...
 2.4.30: Apply Eulers method with successively smaller step sizes on the int...
 2.4.31: The general solution of the equation dy dx D .1 C y2/ cos x is y.x/...
Solutions for Chapter 2.4: Numerical Approximation: Eulers Method
Full solutions for Differential Equations and Boundary Value Problems: Computing and Modeling  5th Edition
ISBN: 9780321796981
Solutions for Chapter 2.4: Numerical Approximation: Eulers Method
Get Full SolutionsSince 31 problems in chapter 2.4: Numerical Approximation: Eulers Method have been answered, more than 15581 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Differential Equations and Boundary Value Problems: Computing and Modeling, edition: 5. Chapter 2.4: Numerical Approximation: Eulers Method includes 31 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Differential Equations and Boundary Value Problems: Computing and Modeling was written by and is associated to the ISBN: 9780321796981.

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).

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

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).

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

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.

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.

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

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

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.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Rank r (A)
= number of pivots = dimension of column space = dimension of row 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 ().

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

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

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

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