 9.7.1: In each of 1 through 6 an autonomous system is expressed in polar c...
 9.7.2: In each of 1 through 6 an autonomous system is expressed in polar c...
 9.7.3: In each of 1 through 6 an autonomous system is expressed in polar c...
 9.7.4: In each of 1 through 6 an autonomous system is expressed in polar c...
 9.7.5: In each of 1 through 6 an autonomous system is expressed in polar c...
 9.7.6: In each of 1 through 6 an autonomous system is expressed in polar c...
 9.7.7: If x = r cos , y = r sin , show that y(dx/dt) x(dy/dt) = r2(d/dt)
 9.7.8: (a) Show that the system dx/dt = y + xf(r)/r, dy/dt = x + yf(r)/r h...
 9.7.9: Determine the periodic solutions, if any, of the system dx dt = y +...
 9.7.10: Using Theorem 9.7.2, show that the linear autonomous system dx/dt =...
 9.7.11: In each of 11 and 12 show that the given system has no periodic sol...
 9.7.12: In each of 11 and 12 show that the given system has no periodic sol...
 9.7.13: Prove Theorem 9.7.2 by completing the following argument. According...
 9.7.14: (a) By examining the graphs of u versus t in Figures 9.7.3, 9.7.5, ...
 9.7.15: The equation u (1 1 3u2 )u + u = 0 is often called the Rayleigh13 e...
 9.7.16: Consider the system of equations x = x + y x(x2 + y2 ), y = x + y y...
 9.7.17: Consider the van der Pol system x = y, y = x + (1 x2 )y, where now ...
 9.7.18: 18 and 19 extend the consideration of the RosenzweigMacArthur preda...
 9.7.19: 18 and 19 extend the consideration of the RosenzweigMacArthur preda...
 9.7.20: There are certain chemical reactions in which the constituent conce...
 9.7.21: The system x = 3(x + y 1 3 x3 k), y = 1 3 (x + 0.8y 0.7) is a speci...
Solutions for Chapter 9.7: Periodic Solutions and Limit Cycles
Full solutions for Elementary Differential Equations and Boundary Value Problems  9th Edition
ISBN: 9780470383346
Solutions for Chapter 9.7: Periodic Solutions and Limit Cycles
Get Full SolutionsChapter 9.7: Periodic Solutions and Limit Cycles includes 21 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 9. Since 21 problems in chapter 9.7: Periodic Solutions and Limit Cycles have been answered, more than 13210 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470383346.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

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!

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

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.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

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].

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Lucas numbers
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.

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.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

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.

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).

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).