 2.1.37: Separate variables and use partial fractions to solve the initial v...
 2.1.38: Separate variables and use partial fractions to solve the initial v...
 2.1.39: Separate variables and use partial fractions to solve the initial v...
 2.1.40: Separate variables and use partial fractions to solve the initial v...
 2.1.41: Separate variables and use partial fractions to solve the initial v...
 2.1.42: Separate variables and use partial fractions to solve the initial v...
 2.1.43: Separate variables and use partial fractions to solve the initial v...
 2.1.44: Separate variables and use partial fractions to solve the initial v...
 2.1.45: The time rate of change of a rabbit population P is proportional to...
 2.1.46: Suppose that the fish population P .t / in a lake is attacked by a ...
 2.1.47: Suppose that when a certain lake is stocked with fish, the birth an...
 2.1.48: The time rate of change of an alligator population P in a swamp is ...
 2.1.49: The time rate of change of an alligator population P in a swamp is ...
 2.1.50: Repeat part (a) of in the case <. What now happens to the rabbit po...
 2.1.51: . Consider a population P .t / satisfying the logistic equation dP=...
 2.1.52: Consider a rabbit population P .t / satisfying the logistic equatio...
 2.1.53: Consider a rabbit population P .t / satisfying the logistic equatio...
 2.1.54: . Consider a population P .t / satisfying the extinctionexplosion e...
 2.1.55: Consider an alligator population P .t / satisfying the extinctione...
 2.1.56: Consider an alligator population P .t / satisfying the extinctione...
 2.1.57: Suppose that the population P .t / of a country satisfies the diffe...
 2.1.58: Suppose that at time t D 0, half of a logistic population of 100;00...
 2.1.59: As the salt KNO3 dissolves in methanol, the number x.t / of grams o...
 2.1.60: Suppose that a community contains 15,000 people who are susceptible...
 2.1.61: The data in the table in Fig. 2.1.7 are given for a certain populat...
 2.1.62: A population P .t / of small rodents has birth rate D .0:001/P (bir...
 2.1.63: Consider an animal population P .t / with constant death rate D 0:0...
 2.1.64: Suppose that the number x.t / (with t in months) of alligators in a...
 2.1.65: During the period from 1790 to 1930, the U.S. population P .t / (t ...
 2.1.66: A tumor may be regarded as a population of multiplying cells. It is...
 2.1.67: For the tumor of 30, suppose that at time t D 0 there are P0 D 106 ...
 2.1.68: Derive the solution P .t / D MP0 P0 C .M P0/ekM t of the logistic i...
 2.1.69: (a) Derive the solution P .t / D MP0 P0 C .M P0/ekM t of the extinc...
 2.1.70: If P .t / satisfies the logistic equation in (3), use the chain rul...
 2.1.71: Consider two population functions P1.t / and P2.t /, both of which ...
 2.1.72: To solve the two equations in (10) for the values of k and M, begin...
 2.1.73: Use the method of to fit the logistic equation to the actual U.S. p...
 2.1.74: Fit the logistic equation to the actual U.S. population data (Fig. ...
 2.1.75: Birth and death rates of animal populations typically are not const...
Solutions for Chapter 2.1: Population Models
Full solutions for Differential Equations and Boundary Value Problems: Computing and Modeling  5th Edition
ISBN: 9780321796981
Solutions for Chapter 2.1: Population Models
Get Full SolutionsThis textbook survival guide was created for the textbook: Differential Equations and Boundary Value Problems: Computing and Modeling, edition: 5. This expansive textbook survival guide covers the following chapters and their solutions. Since 39 problems in chapter 2.1: Population Models have been answered, more than 16399 students have viewed full stepbystep solutions from this chapter. Differential Equations and Boundary Value Problems: Computing and Modeling was written by and is associated to the ISBN: 9780321796981. Chapter 2.1: Population Models includes 39 full stepbystep solutions.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

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.

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!).

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

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.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

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

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

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.