 9.2.9.1.22: Use the RK4 method with h 0.1 to approximate y(0.5), where y(x) is ...
 9.2.9.1.23: Assume that in (4). Use the resulting secondorder RungeKutta metho...
 9.2.9.1.24: In 312 use the RK4 method with h 0.1 to obtain a fourdecimal appro...
 9.2.9.1.25: In 312 use the RK4 method with h 0.1 to obtain a fourdecimal appro...
 9.2.9.1.26: In 312 use the RK4 method with h 0.1 to obtain a fourdecimal appro...
 9.2.9.1.27: In 312 use the RK4 method with h 0.1 to obtain a fourdecimal appro...
 9.2.9.1.28: In 312 use the RK4 method with h 0.1 to obtain a fourdecimal appro...
 9.2.9.1.29: In 312 use the RK4 method with h 0.1 to obtain a fourdecimal appro...
 9.2.9.1.30: In 312 use the RK4 method with h 0.1 to obtain a fourdecimal appro...
 9.2.9.1.31: In 312 use the RK4 method with h 0.1 to obtain a fourdecimal appro...
 9.2.9.1.32: In 312 use the RK4 method with h 0.1 to obtain a fourdecimal appro...
 9.2.9.1.33: In 312 use the RK4 method with h 0.1 to obtain a fourdecimal appro...
 9.2.9.1.34: If air resistance is proportional to the square of the instantaneou...
 9.2.9.1.35: A mathematical model for the area A (in cm2) that a colony of bacte...
 9.2.9.1.36: Consider the initialvalue problem y x2 y3 , y(1) 1. See in Exercis...
 9.2.9.1.37: Consider the initialvalue problem y 2y, y(0) 1. The analytic solut...
 9.2.9.1.38: Repeat using the initialvalue problem y2y x, y(0) 1. The analytic ...
 9.2.9.1.39: Consider the initialvalue problem y 2x 3y 1, y(1) 5. The analytic ...
 9.2.9.1.40: Repeat for the initialvalue problem y ey , y(0) 0. The analytic so...
 9.2.9.1.41: A count of the number of evaluations of the function f used in solv...
 9.2.9.1.42: The RK4 method for solving an initialvalue problem over an interva...
Solutions for Chapter 9.2: Numerical Solutions of Ordinary Differential Equations
Full solutions for Differential Equations with BoundaryValue Problems,  8th Edition
ISBN: 9781111827069
Solutions for Chapter 9.2: Numerical Solutions of Ordinary Differential Equations
Get Full SolutionsSince 21 problems in chapter 9.2: Numerical Solutions of Ordinary Differential Equations have been answered, more than 23154 students have viewed full stepbystep solutions from this chapter. Chapter 9.2: Numerical Solutions of Ordinary Differential Equations includes 21 full stepbystep solutions. This textbook survival guide was created for the textbook: Differential Equations with BoundaryValue Problems,, edition: 8. Differential Equations with BoundaryValue Problems, was written by and is associated to the ISBN: 9781111827069. This expansive textbook survival guide covers the following chapters and their solutions.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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

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

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

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!).

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

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(DÂ» O.

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.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

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

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.