- 11.4.1: A pendulum is released at u p/3 and is given an initial angular vel...
- 11.4.2: (a) If a pendulum is released from rest at u u0, show that the angu...
- 11.4.3: A bead with mass m slides along a thin wire whose shape is describe...
- 11.4.4: A bead with mass m slides along a thin wire whose shape is describe...
- 11.4.5: A bead is released from the position x(0) x0 on the curve z 1 2x2 w...
- 11.4.6: Rework with z cosh x.
- 11.4.7: Refer to Figure 11.4.9. If xm x1 xM and x x1, show that F(x)G(y) c0...
- 11.4.8: From 1. and 3. on page 655, conclude that the maximum number of pre...
- 11.4.9: In many fishery science models the rate at which a species is caugh...
- 11.4.10: A predatorprey interaction is described by the LotkaVolterra model ...
- 11.4.11: A competitive interaction is described by the LotkaVolterra competi...
- 11.4.12: In (1) show that (0, 0) is always an unstable node.
- 11.4.13: In (1) show that (K1, 0) is a stable node when K1 K2/a21 and a sadd...
- 11.4.14: Use 12 and 13 to establish that (0, 0), (K1, 0), and (0, K2) are un...
- 11.4.15: In (1) show that X (x, y) is a saddle point when K1 a12 , K2 and K2...
- 11.4.16: If we assume that a damping force acts in a direction opposite to t...
- 11.4.17: In the analysis of free, damped motion in Section 3.8 we assumed th...
- 11.4.18: A bead with mass m slides along a thin wire whose shape may be desc...
- 11.4.19: An undamped oscillation satisfies a nonlinear second- order differe...
- 11.4.20: The LotkaVolterra predatorprey model assumes that, in the absence o...
- 11.4.21: The nonlinear system x9 a y 1 y x 2 x y9 y 1 y x 2 y b arises in a ...
- 11.4.22: Use the methods of this chapter together with a numerical solver to...
Solutions for Chapter 11.4: Autonomous Systems as Mathematical Models
Full solutions for Advanced Engineering Mathematics | 6th 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.
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
A = CTC = (L.J]))(L.J]))T for positive definite A.
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
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.
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal 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.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Constant down each diagonal = time-invariant (shift-invariant) filter.
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.