- 7.3.1: dy dt = 2y + 1, y(0) = 3, 0 t 2, n = 4
- 7.3.2: dy dt = t y2, y(0) = 1, 0 t 1, n = 4
- 7.3.3: dy dt = (3 y)(y + 1), y(0) = 0, 0 t 5, n = 10
- 7.3.4: dy dt = e2/y , y(0) = 2, 0 t 2, n = 4
- 7.3.5: dy dt = e2/y , y(1) = 2, 1 t 3, n = 4
- 7.3.6: Consider the initial-value problem dy dt = 2t y2, y(0) = 1 over the...
- 7.3.7: Consider the predator-prey system d R dt = 2R 1.2RF d F dt = F + 1....
- 7.3.8: Use Runge-Kutta to approximate the solution to the second-order equ...
Solutions for Chapter 7.3: THE RUNGE-KUTTA METHOD
Full solutions for Differential Equations 00 | 4th Edition
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.
A = CTC = (L.J]))(L.J]))T for positive definite A.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
Incidence matrix of a directed graph.
The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .
Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
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.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
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.
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
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.
Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.
Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.