 2.5.1: 1 through 4 involve equations of the form dy/dt = f ( y). In each p...
 2.5.2: 1 through 4 involve equations of the form dy/dt = f ( y). In each p...
 2.5.3: 1 through 4 involve equations of the form dy/dt = f ( y). In each p...
 2.5.4: 1 through 4 involve equations of the form dy/dt = f ( y). In each p...
 2.5.5: Semistable Equilibrium Solutions. Sometimes a constant equilibrium ...
 2.5.6: 6 through 9 involve equations of the form dy/dt = f ( y). In each p...
 2.5.7: 6 through 9 involve equations of the form dy/dt = f ( y). In each p...
 2.5.8: 6 through 9 involve equations of the form dy/dt = f ( y). In each p...
 2.5.9: 6 through 9 involve equations of the form dy/dt = f ( y). In each p...
 2.5.10: Complete the derivation of the explicit formula for the solution (1...
 2.5.11: In Example 1, complete the manipulations needed to arrive at equati...
 2.5.12: Complete the derivation of the location of the vertical asymptote i...
 2.5.13: Complete the derivation of formula (18) for the locations of the in...
 2.5.14: Consider the equation dy/dt = f ( y) and suppose that y1 is a criti...
 2.5.15: Suppose that a certain population obeys the logistic equation dy/dt...
 2.5.16: Another equation that has been used to model population growth is t...
 2.5.17: a. Solve the Gompertz equation dy dt = ry ln_K y _, subject to the ...
 2.5.18: A pond forms as water collects in a conical depression of radius a ...
 2.5.19: At a given level of effort, it is reasonable to assume that the rat...
 2.5.20: In this problem we assume that fish are caught at a constant rate h...
 2.5.21: Suppose that a given population can be divided into two parts: thos...
 2.5.22: Some diseases (such as typhoid fever) are spread largely by carrier...
 2.5.23: Daniel Bernoullis work in 1760 had the goal of appraising the effec...
 2.5.24: Consider the equation dy dt = a y2. (29) a. Find all of the critica...
 2.5.25: Consider the equation dy dt = ay y3 = y(a y2). (30) G a. Again cons...
 2.5.26: Consider the equation dy dt = ay y2 = y(a y). (31) a. Again conside...
 2.5.27: Chemical Reactions. A secondorder chemical reaction involves the i...
Solutions for Chapter 2.5: Autonomous Differential Equations and Population Dynamics
Full solutions for Elementary Differential Equations and Boundary Value Problems  11th Edition
ISBN: 9781119256007
Solutions for Chapter 2.5: Autonomous Differential Equations and Population Dynamics
Get Full SolutionsSince 27 problems in chapter 2.5: Autonomous Differential Equations and Population Dynamics have been answered, more than 12669 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 11. Chapter 2.5: Autonomous Differential Equations and Population Dynamics includes 27 full stepbystep solutions. 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: 9781119256007.

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).

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

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

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.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex 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!).

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Inverse matrix AI.
Square matrix with AI A = I and AAl = 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 B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

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.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

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

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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.

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

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.