 6.6.1.1: In I through 10, an initial value problem and its exact solution y(...
 6.6.2.1: A handheld calculator will suffice for 1 through 10, where an init...
 6.6.3.1: A handheld calculator will suffice for 1 through 10, where an init...
 6.6.4.1: A handheld calculator will suffice for 1 through 8. In each proble...
 6.6.1.2: In I through 10, an initial value problem and its exact solution y(...
 6.6.2.2: A handheld calculator will suffice for 1 through 10, where an init...
 6.6.3.2: A handheld calculator will suffice for 1 through 10, where an init...
 6.6.4.2: A handheld calculator will suffice for 1 through 8. In each proble...
 6.6.1.3: In I through 10, an initial value problem and its exact solution y(...
 6.6.2.3: A handheld calculator will suffice for 1 through 10, where an init...
 6.6.3.3: A handheld calculator will suffice for 1 through 10, where an init...
 6.6.4.3: A handheld calculator will suffice for 1 through 8. In each proble...
 6.6.1.4: In I through 10, an initial value problem and its exact solution y(...
 6.6.2.4: A handheld calculator will suffice for 1 through 10, where an init...
 6.6.3.4: A handheld calculator will suffice for 1 through 10, where an init...
 6.6.4.4: A handheld calculator will suffice for 1 through 8. In each proble...
 6.6.1.5: In I through 10, an initial value problem and its exact solution y(...
 6.6.2.5: A handheld calculator will suffice for 1 through 10, where an init...
 6.6.3.5: A handheld calculator will suffice for 1 through 10, where an init...
 6.6.4.5: A handheld calculator will suffice for 1 through 8. In each proble...
 6.6.1.6: In I through 10, an initial value problem and its exact solution y(...
 6.6.2.6: A handheld calculator will suffice for 1 through 10, where an init...
 6.6.3.6: A handheld calculator will suffice for 1 through 10, where an init...
 6.6.4.6: A handheld calculator will suffice for 1 through 8. In each proble...
 6.6.1.7: In I through 10, an initial value problem and its exact solution y(...
 6.6.2.7: A handheld calculator will suffice for 1 through 10, where an init...
 6.6.3.7: A handheld calculator will suffice for 1 through 10, where an init...
 6.6.4.7: A handheld calculator will suffice for 1 through 8. In each proble...
 6.6.1.8: In I through 10, an initial value problem and its exact solution y(...
 6.6.2.8: A handheld calculator will suffice for 1 through 10, where an init...
 6.6.3.8: A handheld calculator will suffice for 1 through 10, where an init...
 6.6.4.8: A handheld calculator will suffice for 1 through 8. In each proble...
 6.6.1.9: In I through 10, an initial value problem and its exact solution y(...
 6.6.2.9: A handheld calculator will suffice for 1 through 10, where an init...
 6.6.3.9: A handheld calculator will suffice for 1 through 10, where an init...
 6.6.4.9: A computer will be requiredfor the remaining problems in this secti...
 6.6.1.10: In I through 10, an initial value problem and its exact solution y(...
 6.6.2.10: A handheld calculator will suffice for 1 through 10, where an init...
 6.6.3.10: A handheld calculator will suffice for 1 through 10, where an init...
 6.6.4.10: A computer will be requiredfor the remaining problems in this secti...
 6.6.1.11: A programmable calculator or a computer will be useful for I I thro...
 6.6.2.11: A programmable calculator or a computer will be useful for 11 throu...
 6.6.3.11: A programmable calculator or a computer will be useful for 11 throu...
 6.6.4.11: A computer will be requiredfor the remaining problems in this secti...
 6.6.1.12: A programmable calculator or a computer will be useful for I I thro...
 6.6.2.12: A programmable calculator or a computer will be useful for 11 throu...
 6.6.3.12: A programmable calculator or a computer will be useful for 11 throu...
 6.6.4.12: A computer will be requiredfor the remaining problems in this secti...
 6.6.1.13: A programmable calculator or a computer will be useful for I I thro...
 6.6.2.13: A programmable calculator or a computer will be useful for 11 throu...
 6.6.3.13: A programmable calculator or a computer will be useful for 11 throu...
 6.6.4.13: Suppose that a crossbow bolt is shot straight upward with initial v...
 6.6.1.14: A programmable calculator or a computer will be useful for I I thro...
 6.6.2.14: A programmable calculator or a computer will be useful for 11 throu...
 6.6.3.14: A programmable calculator or a computer will be useful for 11 throu...
 6.6.4.14: Repeat 13, but assume instead that the deceleration of the bolt due...
 6.6.1.15: A programmable calculator or a computer will be useful for I I thro...
 6.6.2.15: A programmable calculator or a computer will be useful for 11 throu...
 6.6.3.15: A programmable calculator or a computer will be useful for 11 throu...
 6.6.4.15: Suppose that a projectile is fired straight upward with initial vel...
 6.6.1.16: A programmable calculator or a computer will be useful for I I thro...
 6.6.2.16: A programmable calculator or a computer will be useful for 11 throu...
 6.6.3.16: A programmable calculator or a computer will be useful for 11 throu...
 6.6.4.16: 16 through 18 deal with the batted baseball of Example 4, having in...
 6.6.1.17: A computer with a printer is requiredfor 17 through 24. In these in...
 6.6.2.17: A computer with a printer is required for 17 through 24. In these i...
 6.6.3.17: A computer with a printer is required for 17 through 24. In these i...
 6.6.4.17: 16 through 18 deal with the batted baseball of Example 4, having in...
 6.6.1.18: A computer with a printer is requiredfor 17 through 24. In these in...
 6.6.2.18: A computer with a printer is required for 17 through 24. In these i...
 6.6.3.18: A computer with a printer is required for 17 through 24. In these i...
 6.6.4.18: 16 through 18 deal with the batted baseball of Example 4, having in...
 6.6.1.19: A computer with a printer is requiredfor 17 through 24. In these in...
 6.6.2.19: A computer with a printer is required for 17 through 24. In these i...
 6.6.3.19: A computer with a printer is required for 17 through 24. In these i...
 6.6.4.19: Find the initial velocity of a baseball hit by Babe Ruth (with c = ...
 6.6.1.20: A computer with a printer is requiredfor 17 through 24. In these in...
 6.6.2.20: A computer with a printer is required for 17 through 24. In these i...
 6.6.3.20: A computer with a printer is required for 17 through 24. In these i...
 6.6.4.20: Consider the crossbow bolt of 14, fired with the same initial veloc...
 6.6.1.21: A computer with a printer is requiredfor 17 through 24. In these in...
 6.6.2.21: A computer with a printer is required for 17 through 24. In these i...
 6.6.3.21: A computer with a printer is required for 17 through 24. In these i...
 6.6.4.21: Suppose that an artillery projectile is fired from ground level wit...
 6.6.1.22: A computer with a printer is requiredfor 17 through 24. In these in...
 6.6.2.22: A computer with a printer is required for 17 through 24. In these i...
 6.6.3.22: A computer with a printer is required for 17 through 24. In these i...
 6.6.1.23: A computer with a printer is requiredfor 17 through 24. In these in...
 6.6.2.23: A computer with a printer is required for 17 through 24. In these i...
 6.6.3.23: A computer with a printer is required for 17 through 24. In these i...
 6.6.1.24: A computer with a printer is requiredfor 17 through 24. In these in...
 6.6.2.24: A computer with a printer is required for 17 through 24. In these i...
 6.6.3.24: A computer with a printer is required for 17 through 24. In these i...
 6.6.1.25: You bail out of the helicopter of Example 2 and immediately pull th...
 6.6.2.25: As in of Section 6.1, you bail out of a helicopter and immediately ...
 6.6.3.25: As in of Section 6.2, you bail out of a helicopter and immediately ...
 6.6.1.26: Suppose the deer population P (t) in a small forest initially numbe...
 6.6.2.26: As in of Section 6. 1, suppose the deer population P(t) in a small ...
 6.6.3.26: As in of Section 6.2, suppose the deer population P(t) in a small f...
 6.6.1.27: Use Euler's method with a computer system to find the desired solut...
 6.6.2.27: Use the improved Euler method with a computer system to find the de...
 6.6.3.27: Use the RungeKutta method with a computer system to find the desir...
 6.6.1.28: Use Euler's method with a computer system to find the desired solut...
 6.6.2.28: Use the improved Euler method with a computer system to find the de...
 6.6.3.28: Use the RungeKutta method with a computer system to find the desir...
 6.6.1.29: Consider the initial value problem dy 7x +y=0, y(l) = 1 . dx (a) ...
 6.6.2.29: Consider the crossbow bolt of Example 2 in Section 1.8, shot straig...
 6.6.3.29: In 29 and 30, the linear acceleration a = dv/dt of a moving particl...
 6.6.1.30: Apply Euler's method with successively smaller step sizes on the in...
 6.6.2.30: Consider now the crossbow bolt of Example 3 in Section 1.8. It stil...
 6.6.3.30: In 29 and 30, the linear acceleration a = dv/dt of a moving particl...
 6.6.1.31: The general solution of the equation dy = (l + i) cos x dx is y(x) ...
Solutions for Chapter 6: Numerical Methods
Full solutions for Elementary Differential Equations  6th Edition
ISBN: 9780132397308
Solutions for Chapter 6: Numerical Methods
Get Full SolutionsThis textbook survival guide was created for the textbook: Elementary Differential Equations, edition: 6. Since 112 problems in chapter 6: Numerical Methods have been answered, more than 9045 students have viewed full stepbystep solutions from this chapter. Chapter 6: Numerical Methods includes 112 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Elementary Differential Equations was written by and is associated to the ISBN: 9780132397308.

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!

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).

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

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].

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!).

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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.

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.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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.

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.

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.

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

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