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 .
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.
Suppose we find the sum of solutions obtained in (i) and (ii) of (b) above. Does this give a solution of the equation describing the motion? If yes, what physical situation, this solution will be correspond to? When is the velocity of the pendulum maximum? What type of energy does the pendulum have at position of maximum velocity?
A particle is moving under a central force along the path given by the equation r = a tan .Show that the radial acceleration is
where,
A tumour is grown in a laboratory with a plentiful supply of nutrients. The number of cells in the tumour is proportional to its volume V at time t . Its growth rate is then . The growth rate decreases exponentially with t . Its value at time t being found empirically to
, where a and l are positive constants.
i) express this empirical law as a differential equation
ii) find the solution of the differential equation obtain subject to the initial condition V = V0 at t = 0.
A particle is moving under a central attracting force F. Show that its equations of motion can be written as
where
A profit maximizing monopolist produces a quantity of a commodity. His
D(p) = 30 , where p is the price. Find the values of p, q and the profit.