 2.1.1P: Separate variables and use partial fractions to solve the initial v...
 2.1.2P: Separate variables and use partial fractions to solve the initial v...
 2.1.3P: Separate variables and use partial fractions to solve the initial v...
 2.1.4P: Separate variables and use partial fractions to solve the initial v...
 2.1.5P: Separate variables and use partial fractions to solve the initial v...
 2.1.6P: Separate variables and use partial fractions to solve the initial v...
 2.1.7P: Separate variables and use partial fractions to solve the initial v...
 2.1.8P: Separate variables and use partial fractions to solve the initial v...
 2.1.9P: The time rate of change of a rabbit population P is proportional to...
 2.1.10P: Suppose that the fish population P(t) in a lake is attacked by a di...
 2.1.11P: Suppose that when a certain lake is stocked with fish, the birth an...
 2.1.12P: The time rate of change of an alligator population P in a swamp is ...
 2.1.13P: Consider a prolific breed of rabbits whose birth and death rates, ?...
 2.1.14P: Repeat part (a) of the in the case ? < ?. What now happens to the r...
 2.1.15P: Consider a population P(t) satisfying the logistic equation dP/dt =...
 2.1.16P: Consider a rabbit population P(1) satisfying the logistic equation ...
 2.1.17P: Consider a rabbit population P(t) satisfying the logistic equation ...
 2.1.18P: Consider a population P(t) satisfying the extinctionexplosion equa...
 2.1.19P: Consider an alligator population P(t) satisfying the extinction/exp...
 2.1.20P: Consider an alligator population P(t) satisfying the extinction/exp...
 2.1.21P: Suppose that the population P(t) of a country satisfies the differe...
 2.1.22P: Suppose that at time t = 0, half of a “logistic” population of 100,...
 2.1.23P: As the salt KNO3 dissolves in methanol, the number x(t) of grams of...
 2.1.24P: Suppose that a community contains 15,000 people who are susceptible...
 2.1.25P: The data in the table in Fig. are given for a certain population P(...
 2.1.26P: A population P(t) of small rodents has birth rate ? = (0.001) P (bi...
 2.1.27P: Consider an animal population P(t) with constant death rate ? = 0.0...
 2.1.28P: Suppose that the number x(t) (with t in months) of alligators in a ...
 2.1.29P: During the period from 1790 to 1930, the U.S. population P(t) (t in...
 2.1.30P: A tumor may be regarded as a population of multiplying cells. It is...
 2.1.31P: For the tumor of 30, suppose that at time t = 0 there are Po = 106 ...
 2.1.32P: Derive the solution of the logistic initial value problem P? = kP(M...
 2.1.33P: (a) Derive the solution of the extinctionexplosion initial value p...
 2.1.34P: If P(t) satisfies the logistic equation, use the chain rule to show...
 2.1.35P: Consider two population functions P1(t) and P2(t), both of which sa...
 2.1.36P: To solve the two equations in for the values of k and M, begin by s...
 2.1.37P: Use the method of fit the logistic equation to the actual U.S. popu...
 2.1.38P: Fit the logistic equation to the actual U.S. population data for th...
 2.1.39P: Birth and death rales of animal populations typically are not const...
Solutions for Chapter 2.1: Differential Equations and Linear Algebra 3rd Edition
Full solutions for Differential Equations and Linear Algebra  3rd Edition
ISBN: 9780136054252
Solutions for Chapter 2.1
Get Full SolutionsSince 39 problems in chapter 2.1 have been answered, more than 11495 students have viewed full stepbystep solutions from this chapter. Chapter 2.1 includes 39 full stepbystep solutions. Differential Equations and Linear Algebra was written by and is associated to the ISBN: 9780136054252. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Differential Equations and Linear Algebra, edition: 3.

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!

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

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].

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

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

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 .

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

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 •

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.

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).

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

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.

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

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