 8.2.1: In each of 1 through 6 find approximate values of the solution of t...
 8.2.2: In each of 1 through 6 find approximate values of the solution of t...
 8.2.3: In each of 1 through 6 find approximate values of the solution of t...
 8.2.4: In each of 1 through 6 find approximate values of the solution of t...
 8.2.5: In each of 1 through 6 find approximate values of the solution of t...
 8.2.6: In each of 1 through 6 find approximate values of the solution of t...
 8.2.7: In each of 7 through 12 find approximate values of the solution of ...
 8.2.8: In each of 7 through 12 find approximate values of the solution of ...
 8.2.9: In each of 7 through 12 find approximate values of the solution of ...
 8.2.10: In each of 7 through 12 find approximate values of the solution of ...
 8.2.11: In each of 7 through 12 find approximate values of the solution of ...
 8.2.12: In each of 7 through 12 find approximate values of the solution of ...
 8.2.13: Complete the calculations leading to the entries in columns four an...
 8.2.14: In this problem we establish that the local truncation error for th...
 8.2.15: Consider the improved Euler method for solving the illustrative ini...
 8.2.16: In each of 16 and 17 use the actual solution (t) to determine en+1 ...
 8.2.17: In each of 16 and 17 use the actual solution (t) to determine en+1 ...
 8.2.18: In each of 18 through 21 carry out one step of the Euler method and...
 8.2.19: In each of 18 through 21 carry out one step of the Euler method and...
 8.2.20: In each of 18 through 21 carry out one step of the Euler method and...
 8.2.21: In each of 18 through 21 carry out one step of the Euler method and...
 8.2.22: The modified Euler formula for the initial value problem y = f(t, y...
 8.2.23: In each of 23 through 26 use the modified Euler formula of with h =...
 8.2.24: In each of 23 through 26 use the modified Euler formula of with h =...
 8.2.25: In each of 23 through 26 use the modified Euler formula of with h =...
 8.2.26: In each of 23 through 26 use the modified Euler formula of with h =...
Solutions for Chapter 8.2: Improvements on the Euler Method
Full solutions for Elementary Differential Equations and Boundary Value Problems  9th Edition
ISBN: 9780470383346
Solutions for Chapter 8.2: Improvements on the Euler Method
Get Full SolutionsSince 26 problems in chapter 8.2: Improvements on the Euler Method have been answered, more than 13209 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 8.2: Improvements on the Euler Method includes 26 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 9. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470383346.

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

Column space C (A) =
space of all combinations of the columns of A.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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 II· Condition numbers measure the sensitivity of the output to change in the input.

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).

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.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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.

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.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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.

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).

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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

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).

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

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