 2.1.1: In one of these systems, the prey are very large animals and the pr...
 2.1.2: Find all equilibrium points for the two systems. Explain the signif...
 2.1.3: Suppose that the predators are extinct at time t0 = 0. For each sys...
 2.1.4: For each system, describe the behavior of the prey population if th...
 2.1.5: For each system, suppose that the prey are extinct at time t0 = 0. ...
 2.1.6: For each system, describe the behavior of the predator population i...
 2.1.7: Consider the predatorprey system d R dt = 2 1 R 3 R RF d F dt = 2F...
 2.1.8: Consider the predatorprey system d R dt = 2R 1 R 2.5 1.5RF d F dt ...
 2.1.9: How would you modify these systems to include the effect of hunting...
 2.1.10: How would you modify these systems to include the effect of hunting...
 2.1.11: Suppose the predators discover a second, unlimited source of food, ...
 2.1.12: Suppose the predators found a second food source that is limited in...
 2.1.13: Suppose predators migrate to an area if there are five times as man...
 2.1.14: Suppose prey move out of an area at a rate proportional to the numb...
 2.1.15: Consider the two systems of differential equations (i) dx dt = 0.3x...
 2.1.16: Consider the system of predatorprey equations d R dt = 2 1 R 3 R R...
 2.1.17: Pesticides that kill all insect species are not only bad for the en...
 2.1.18: Some predator species seldom capture healthy adult prey, eating onl...
 2.1.19: Consider the initialvalue problem d2 y dt2 + y = 0 with y(0) = 0 a...
 2.1.20: Consider the equation d2 y dt2 + k m y = 0 for the motion of a simp...
 2.1.21: A mass weighing 12 pounds stretches a spring 3 inches. What is the ...
 2.1.22: A mass weighing 4 pounds stretches a spring 4 inches. (a) Formulate...
 2.1.23: Do the springs in an extra firm mattress have a large spring consta...
 2.1.24: Consider a vertical massspring system as shown in the figure below...
 2.1.25: Write a system of differential equations that models the evolution ...
 2.1.26: Describe an experiment you could perform to determine an approximat...
 2.1.27: Suppose substances A and B are added to the solution at constant (p...
 2.1.28: Suppose A and B are being added to the solution at constant (perhap...
 2.1.29: Suppose A and B are being added to the solutions at constant (perha...
 2.1.30: Suppose A and B are being added to the solution at constant (perhap...
Solutions for Chapter 2.1: MODELING VIA SYSTEMS
Full solutions for Differential Equations 00  4th Edition
ISBN: 9780495561989
Solutions for Chapter 2.1: MODELING VIA SYSTEMS
Get Full SolutionsChapter 2.1: MODELING VIA SYSTEMS includes 30 full stepbystep solutions. Differential Equations 00 was written by and is associated to the ISBN: 9780495561989. Since 30 problems in chapter 2.1: MODELING VIA SYSTEMS have been answered, more than 16317 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Differential Equations 00, edition: 4. This expansive textbook survival guide covers the following chapters and their solutions.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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

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.

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

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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.

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.

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 .

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

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

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

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

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

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.