Consider the following first-order ODE formulation = a{L- n(t)}, n(t0) = n0
.
Associate the variables {t, n(t)} and the parameters {a, L} to a physical system so that the above formulation becomes a mathematical model for population changes.
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Consider the population x(t) of fish and y(t) of sharks at any time t in a certain region of the pacific ocean. Sharks prey on fishes and their population would decrease if isolated from the fish population. Let us assume further the following assumptions.
1) The change in the shark and fish population, in isolation, is respectively proportional to the present population of sharks and fish.
2) The number of sharks and fish caught by fisherman is directly proportional to the present population of the shark and fish population respectively. The fishing methods do not discriminate between sharks or fish.
3) The number of fish eaten by sharks is directly proportional to the product of the number of fish present and the number of sharks present.
4) The additional number of sharks surviving is directly proportional to the number of fish eaten. Under assumptions 1) to 4)
i) Formulate the mathematical model for the given problem and write a
system of differential equations governing it.
ii) Is the system of equations obtained in i) an autonomous system? Give reasons.
iii) Find the steady-state solution of the system in i) above. What does the solutions obtained correspond to?
iv) From the system of equations obtained in i) obtain the expression for
dy / dx and hence solve it under the condition x(0) = x0 , y(0) = y0 .
Television sets for repair arrive at random at an average of 4 per day to a single repairman
who takes an average of hours to carry out each repair. It being assumed that the repair times have an exponential distribution. What is the average number of television sets in the workshop? What is the probability that an arriving set will find at least 3 sets in front of it?
The repairman works for eight hours a day.
Consider the following first-order ODE formulation = a{L- n(t)}, n(t0) = n0
.
Associate the variables {t, n(t)} and the parameters {a, L} to a physical system so that the above formulation becomes a mathematical model for population changes.
In a perfectly competitive market, the supply function S(p) and demand function D(p) are given as follows:
, where p is the price:
i) Find the equilibrium prices
ii) Using Walrasian stability condition, determine whether the equilibrium prices are stable.
A particle is moving under a central attracting force F. Show that its equations of motion can be written as
where
The process of plucking of a guitar string results in the frequency of vibration being a function of the young’s modulus, the length and density of the string. Use dimensional analysis to derive an expression for the frequency of vibration of the string.
For the problem of finding the period of oscillation of a simple pendulum, obtain a formulation in the case when air resistance is proportional to the square of the velocity. Find a solution for this formulation.