- 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 second-order 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
Solutions for Chapter 2.5: Autonomous Differential Equations and Population DynamicsGet Full Solutions
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.
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.
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
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 Fn-1c 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!).
Gram-Schmidt 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 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).
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)j-I and det V = product of (Xk - Xi) for k > i.