 2.8.1: In each of 1 and 2 transform the given initial value problem into a...
 2.8.2: In each of 1 and 2 transform the given initial value problem into a...
 2.8.3: In each of 3 through 6 let 0(t) = 0 and use the method of successiv...
 2.8.4: In each of 3 through 6 let 0(t) = 0 and use the method of successiv...
 2.8.5: In each of 3 through 6 let 0(t) = 0 and use the method of successiv...
 2.8.6: In each of 3 through 6 let 0(t) = 0 and use the method of successiv...
 2.8.7: In each of 7 and 8 let 0(t) = 0 and use the method of successive ap...
 2.8.8: In each of 7 and 8 let 0(t) = 0 and use the method of successive ap...
 2.8.9: In each of 9 and 10 let 0(t) = 0 and use the method of successive a...
 2.8.10: In each of 9 and 10 let 0(t) = 0 and use the method of successive a...
 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 that limn n(x) = 0, 0 x < 1, 1, x ...
 2.8.14: Let n(x) = xn for 0 x 1 and show that limn n(x) = 0, 0 x < 1, 1, 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  9th Edition
ISBN: 9780470383346
Solutions for Chapter 2.8: The Existence and Uniqueness Theorem
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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!

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib IIĀ· Condition numbers measure the sensitivity of the output to change in the input.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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

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.

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

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

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

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

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.

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.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).