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MATH 33B Lecture Notes - Lecture 5: The Mixtures

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rubyporcupine828
17 Nov 2016
School
UCLA
Department
Mathematics
Course
MATH 33B
Professor
Visan

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Document Summary

Assume (cid:2869)(cid:4666)(cid:4667) and (cid:2870)(cid:4666)(cid:4667) are solutions to (cid:4666)(cid:884)(cid:4667), that is, (cid:4666)(cid:4667)=(cid:1857)(cid:4666)(cid:4667) where (cid:4666) , (cid:4667) constant and (cid:4666)(cid:4667)=(cid:1858)(cid:4666)(cid:4667) F(cid:396)o(cid:373) the fo(cid:396)(cid:373)ula, it"s (cid:272)lea(cid:396) that solutio(cid:374)s to the ho(cid:373)oge(cid:374)eous e(cid:395)uatio(cid:374) diffe(cid:396) (cid:271)y a multiplicative constant. Consider (cid:1856)(cid:1856)=(cid:1858)(cid:4666)(cid:4667) (cid:4666)(cid:883)(cid:4667) (cid:1856)(cid:1856)=(cid:1858)(cid:4666)(cid:4667)+(cid:1859)(cid:4666)(cid:4667) (cid:4666)(cid:884)(cid:4667) (cid:1856)(cid:2869)(cid:1856) =(cid:1858)(cid:4666)(cid:4667)(cid:2869)+(cid:1859)(cid:4666)(cid:4667) (cid:1856)(cid:2870)(cid:1856) =(cid:1858)(cid:4666)(cid:4667)(cid:2870)+(cid:1859)(cid:4666)(cid:4667) (cid:1856)(cid:2869)(cid:1856) (cid:1856)(cid:2870)(cid:1856) =(cid:1858)(cid:4666)(cid:4667)(cid:4666)(cid:2869) (cid:2870)(cid:4667) (cid:1856)(cid:1856)(cid:4666)(cid:2869) (cid:2870)(cid:4667)=(cid:1858)(cid:4666)(cid:4667)(cid:4666)(cid:2869) (cid:2870)(cid:4667) So (cid:4666)(cid:2869) (cid:2870)(cid:4667) is a solution to the homogeneous equation (cid:4666)(cid:883)(cid:4667). If (cid:3042) is a solution to the homogeneous equation (cid:4666)(cid:883)(cid:4667) and (cid:3043) is a particular solution to the inhomogeneous equation (cid:4666)(cid:884)(cid:4667), then every solution of the inhomogeneous equation (cid:4666)(cid:884)(cid:4667) takes the form (cid:4666)(cid:4667)=(cid:3043)(cid:4666)(cid:4667)+ (cid:3042)(cid:4666)(cid:4667) for some constant (cid:4666) , (cid:4667). A tank contains 100 gallons of fresh water. Water containing pounds of salt per gallon enters the tank at a rate of 2 gallons per minute. The mixtures leaves the tank through a drain at the bottom of the tank at the rate of 2 gallons per minute. This process stops after 10 minutes, when the following process starts:

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A classical model in mathematical ecology is the Lotkaรขย€ย“Volterra predator-prey model. Consider a simple ecosystem consisting of rabbits that have an infinite supply of food and foxes that prey on the rabbits for their food. This is modeled by a pair of nonlinear, first-order differential equations:

where t is time, r(t) is the number of rabbits, f(t) is the number of foxes, and รŽยฑ is a positive constant. If รŽยฑ = 0, the two populations do not interact, the rabbits do what rabbits do best, and the foxes die off from starvation. If รŽยฑ > 0, the foxes encounter the rabbits with a probability that is proportional to the product of their numbers. Such an encounter results in a decrease in the number of rabbits and (for less obvious reasons) an increase in the number of foxes. The solutions to this nonlinear system cannot be expressed in terms of other known functions; the equations must be solved numerically. It turns out that the solutions are always periodic, with a period that depends on the initial conditions. In other words, for any r(0) and f(0), there is a value t = tp when both populations return to their original values. Consequently, for all t,

(a) Compute the solution with r0 = 300, f0 = 150, and รŽยฑ = 0.01. You should find that tp is close to 5. Make two plots, one of r and f as functions of t and one a phase plane plot with r as one axis and f as the other.

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Hi, I need the full Matlab code and answers for all parts of the problem. Thanks!