 8.1.1: In each of 1 through 6, find approximate values of the solution of ...
 8.1.2: In each of 1 through 6, find approximate values of the solution of ...
 8.1.3: In each of 1 through 6, find approximate values of the solution of ...
 8.1.4: In each of 1 through 6, find approximate values of the solution of ...
 8.1.5: In each of 1 through 6, find approximate values of the solution of ...
 8.1.6: In each of 1 through 6, find approximate values of the solution of ...
 8.1.7: In each of 7 through 12, find approximate values of the solution of...
 8.1.8: In each of 7 through 12, find approximate values of the solution of...
 8.1.9: In each of 7 through 12, find approximate values of the solution of...
 8.1.10: In each of 7 through 12, find approximate values of the solution of...
 8.1.11: In each of 7 through 12, find approximate values of the solution of...
 8.1.12: In each of 7 through 12, find approximate values of the solution of...
 8.1.13: Complete the calculations leading to the entries in columns three a...
 8.1.14: Complete the calculations leading to the entries in columns three a...
 8.1.15: Using three terms in the Taylor series given in Eq. (12) and taking...
 8.1.16: In each of 16 and 17, estimate the local truncation error for the E...
 8.1.17: In each of 16 and 17, estimate the local truncation error for the E...
 8.1.18: In each of 18 through 21, obtain a formula for the local truncation...
 8.1.19: In each of 18 through 21, obtain a formula for the local truncation...
 8.1.20: In each of 18 through 21, obtain a formula for the local truncation...
 8.1.21: In each of 18 through 21, obtain a formula for the local truncation...
 8.1.22: Consider the initial value problemy = cos 5t, y(0) = 1.(a) Determin...
 8.1.23: In this problem we discuss the global truncation error associated w...
 8.1.24: Derive an expression analogous to Eq. (22) for the local truncation...
 8.1.25: Using a step size h = 0.05 and the Euler method, but retaining only...
 8.1.26: The following problem illustrates a danger that occurs because of r...
 8.1.27: The distributive law a(b c) = ab ac does not hold, in general, if t...
Solutions for Chapter 8.1: The Euler or Tangent Line Method
Full solutions for Elementary Differential Equations  10th Edition
ISBN: 9780470458327
Solutions for Chapter 8.1: The Euler or Tangent Line Method
Get Full SolutionsSince 27 problems in chapter 8.1: The Euler or Tangent Line Method have been answered, more than 11638 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 8.1: The Euler or Tangent Line Method includes 27 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Differential Equations, edition: 10. Elementary Differential Equations was written by and is associated to the ISBN: 9780470458327.

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.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

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

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.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

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

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

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].

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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

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

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

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.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

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