 11.5.1: In 18, show that the given plane autonomous system (or secondorder...
 11.5.2: In 18, show that the given plane autonomous system (or secondorder...
 11.5.3: In 18, show that the given plane autonomous system (or secondorder...
 11.5.4: In 18, show that the given plane autonomous system (or secondorder...
 11.5.5: In 18, show that the given plane autonomous system (or secondorder...
 11.5.6: In 18, show that the given plane autonomous system (or secondorder...
 11.5.7: In 18, show that the given plane autonomous system (or secondorder...
 11.5.8: In 18, show that the given plane autonomous system (or secondorder...
 11.5.9: In 9 and 10, use the Dulac negative criterion to show that the give...
 11.5.10: In 9 and 10, use the Dulac negative criterion to show that the give...
 11.5.11: Show that the plane autonomous system x x(1 x2 3y2 ) y y(3 x2 3y2 )...
 11.5.12: If g/x 0 in a region R, prove that x g(x, x) has no periodic soluti...
 11.5.13: Show that the predatorprey model x9 ax bxy y9 cxy r K y(K 2 y) in o...
 11.5.14: In 14 and 15, find a circular invariant region for the given plane ...
 11.5.15: In 14 and 15, find a circular invariant region for the given plane ...
 11.5.16: Verify that the region bounded by the closed curve x6 3y2 1 is an i...
 11.5.17: The plane autonomous system in Example 8 has only one critical poin...
 11.5.18: Use the PoincarBendixson theorem to show that the secondorder nonli...
 11.5.19: Let X X(t) be the solution of the plane autonomous system x y y x (...
 11.5.20: Investigate global stability for the system x y x y x y3 .
 11.5.21: Empirical evidence suggests that the plane autonomous system x x2 y...
 11.5.22: (a) Find and classify all critical points of the plane autonomous s...
Solutions for Chapter 11.5: Periodic Solutions, Limit Cycles, and Global Stability
Full solutions for Advanced Engineering Mathematics  6th Edition
ISBN: 9781284105902
Solutions for Chapter 11.5: Periodic Solutions, Limit Cycles, and Global Stability
Get Full SolutionsAdvanced Engineering Mathematics was written by and is associated to the ISBN: 9781284105902. Chapter 11.5: Periodic Solutions, Limit Cycles, and Global Stability includes 22 full stepbystep solutions. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics , edition: 6. This expansive textbook survival guide covers the following chapters and their solutions. Since 22 problems in chapter 11.5: Periodic Solutions, Limit Cycles, and Global Stability have been answered, more than 35409 students have viewed full stepbystep solutions from this chapter.

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.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (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

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!).

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.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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

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.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

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.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

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

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.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

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