 2.8.1: In each of 1 and 2, transform the given initial value problem into ...
 2.8.2: In each of 1 and 2, transform the given initial value problem into ...
 2.8.3: In each of 3 through 6, let 0(t) = 0 and define {n(t)} by the metho...
 2.8.4: In each of 3 through 6, let 0(t) = 0 and define {n(t)} by the metho...
 2.8.5: In each of 3 through 6, let 0(t) = 0 and define {n(t)} by the metho...
 2.8.6: In each of 3 through 6, let 0(t) = 0 and define {n(t)} by the metho...
 2.8.7: In each of 7 and 8, let 0(t) = 0 and use the method of successive a...
 2.8.8: In each of 7 and 8, let 0(t) = 0 and use the method of successive a...
 2.8.9: In each of 9 and 10, let 0(t) = 0 and use the method of successive ...
 2.8.10: In each of 9 and 10, let 0(t) = 0 and use the method of successive ...
 2.8.11: In each of 11 and 12, let 0(t) = 0 and use the method of successive...
 2.8.12: In each of 11 and 12, let 0(t) = 0 and use the method of successive...
 2.8.13: Let n(x) = xn for 0 x 1 and show thatlimn n(x) =0, 0 x < 1,1, x = 1...
 2.8.14: Consider the sequence n(x) = 2nxenx2, 0 x 1.(a) Show that limn n(x)...
 2.8.15: In 15 through 18, we indicate how to prove that the sequence {n(t)}...
 2.8.16: In 15 through 18, we indicate how to prove that the sequence {n(t)}...
 2.8.17: In 15 through 18, we indicate how to prove that the sequence {n(t)}...
 2.8.18: In 15 through 18, we indicate how to prove that the sequence {n(t)}...
 2.8.19: In this problem we deal with the question of uniqueness of the solu...
Solutions for Chapter 2.8: The Existence and Uniqueness Theorem
Full solutions for Elementary Differential Equations and Boundary Value Problems  10th Edition
ISBN: 9780470458310
Solutions for Chapter 2.8: The Existence and Uniqueness Theorem
Get Full SolutionsChapter 2.8: The Existence and Uniqueness Theorem includes 19 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 19 problems in chapter 2.8: The Existence and Uniqueness Theorem have been answered, more than 17600 students have viewed full stepbystep solutions from this chapter. 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.

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.

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

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.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

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.

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

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

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 one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

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.

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

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·

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

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