Problem #3:A vector y[R(t) F(t)] describes the populationsof some rabbits R(t) and foxes F(t). The populationsobey the system of differential equations given by y'Ay where=A =[=-84-6130593The rabbit population begins at 88500. If we wantthe rabbit population to grow as a simpleexponential of the form R(t) Roest with no otherProblem #3:=terms, how many foxes are needed at time t = 0?(Note that the eigenvalues of A are λ = 3 and 6.)

Given matrix A=-841305-693 The rabbit population R0=88500 Let the population of rabbit be Rt…

Answered: A vector y [R(t) F(t)] describes the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Suppose in an autocatalytic chemical reaction, compound A reacts to form compound B. Next, suppose that the initial concentration of A is CA0 and CB(t) is the concentration of B at time t. Then CA0 CB(t) is the concentration of A at time t. Determine CB(t) if CB(0) = CB0 Note: in an autocatalytic reaction, the resulting substance can stimulate the reaction so that the CB value is proportional to CB(t) and CA0 -CB(t), whose relationship can be written as:dCB(t)/dt= kCB (t)[CA0 - CB(t)], k = reaction rate coefficient dt
According to Newton’s Law of Cooling, the rate of change of the temperature of an objectcan be modeled using the following differential equation: dT/dt=−κ(T−A) where T is the temperature of the object (in◦F), A is the room temperature (in◦F), and κ is the constant of proportionality. (a) Use an appropriate method to solve the differential equation.Remember that κ and A are constants. (b) Use your answer to (a) to solve the following: On a crime show, a dead body is discovered in a hotel room. A forensic technician recordedthat the victim’s body temperature was 91.4◦F at 6:00 pm. One hour later, the coroner arrived and found that the victim’s body temperature had fallen to 88.7◦F. If the thermostatin the hotel room is set at 68◦F, determine when the victim was murdered.(Assume the victim was a healthy 98.6◦F at the time of death.)
Suppose thay a drug is administered to a person in a single dose, and assume that the drug does not accumulate in body tissue, but is excerted through urine. Denote the amount of drug in the body at time t by x1(t) and in the urine at time t by x2(t). If x1(0)=6 mg and x2(0)=0, can you find a system of differential equations for x1(t) and x2(t) if it takes 20 minutes for the drug to be at one-half of its initial amount in the body?
Find a linear differential equation that produces yn (t) = e²t and y(t) = 5(est - 1).
Consider the logistic equation (a) Find the solution satisfying y₁ (0) = 8 and y/₂(0) = -5. yi(t) = 8 Y2(t) (b) Find the time t when y(t) = 4. t= (c) When does y2 (t) become infinite? t=
Differential equations have been used extensively in the study of drug dissolution for patients given oral medications. The three simplest equations used are the zero-order kinetic equation, the Noyes-Whitney equation, and the Weibull equation. All assume that the initial concentration is zero but make different assumptions about how the concentration increases over time during the dissolution of the medication. The zero-order kinetic equation states that the rate of change in the concentration of drug c (in mg/mL) during dissolution is governed by the differential equation dc dt = k where k is a positive constant. Is this differential equation pure-time, autonomous, or nonautonomous? pure time autonomous nonautonomous State in words what this differential equation says about how drug dissolution occurs. The drug concentration increases linearly with time during dissolution. The drug concentration remains constant with time during dissolution. The drug concentration…
Consider a system of differential equations describing the progress of a disease in a population, given by In our particular case, this is: a) Find the nullclines (simplest form) of this system of differential equations. The x-nullcline is y = x = 3-3xy - 1x y = 3xy - 2y where x (t) is the number of susceptible individuals at time t and y(t) is the number of infected individuals at time t. The number of individuals is counted in units of 1,000 individuals. The y-nullclines are y = and x = b) There are two equilibrium points that are biologically meaningful (i.e. whose coordinates are non-negative). They are (x1,9₁) and (x2, y2), where we order them so that x1 < x2. The equilibrium with the smaller x-coordinate is (x₁, y₁) : The equilibrium with the larger x-coordinate is (x2, y₂) = where A = 0 sin (a) c) The linearization of the system of differential equations at the equilibrium (ï1, Y₁) gives a system of the form (+) ¹ (~), f əx ∞ a Ω = A x' G E = F(x, y) for a vector-valued function…

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