A simple model including the seasonal change that affects the growth rate of a population is given by where C is a constant. If 0 x is the initial population, solve the equation and determine the maximum and minimum population.
To solve the given differential equation \(\frac{dx}{dt} = cx(t) \cos t\), we can use separation of variables and integrate both sides:
\[ \int \frac{1}{x} \,dx = \int c \cos t \,dt \]
This leads to:
\[ \ln|x| = c \int \cos t \,dt \]
Integrating the right side:
\[ \ln|x| = c \sin t + K \]
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The yearly fluctuations in the groundwater table are believed to be dependent on the annual rainfall and the volume of water pumped out from the basin. The data collected on these variables for four consecutive years is given below:
Use the method of least squares to find a linear regression equation that best fits the data.
A locality is served by two malls. Each mall has two counters to serve the customers. Both the malls are equally popular and are known to have equal shares of the market. This is evident from the fact that customer’s arrive at each mall’s serving counter at the rate of 12 customers per hour. The average time to serve a customer is 05 minutes. Customers’ arrival is according to a Poisson distribution and the service time is exponential. To provide better service to the customers, the owners of the two malls decide to consolidate into a single larger mall. What is the effect of consolidation on the waiting time of customers?
Do the stability analysis of the trivial equilibrium solution of the following competing species model:
where D1 and D2 are diffusion coefficients of the two population densities N1 and N , 2 respectively. 1 a is the growth rate, 1 b is the predation rate, 1 d is the death rate and 1 c is the conversion rate. The initial boundary conditions are
where Ni are the equilibrium solutions of the given system of equations.
The daily capacity of each of the factory is 150 per day and the daily requirement over each target store is 200. Find the allocation for each factory to each target store which minimize the total transport cost.
The transportation cost of 600 tons of a certain type of material from four factories B1, B2 , B3 , and B4 to three target stores T1,T2 , and T3 are given in the following table:
Formulate the model for which the reproductive function of the cancer cells in the tumor surface is given by 2 1 ; c 1(2 c)2 3 c2 )c( ≠ − − φ = together with initial conditions 5 c = 20×10 at t = .0 Also find the density of the cancer cells in the tumour’s surface area at t = 20 days.
The return distribution on the two securities X and Y are given in the table below:
| Possible Rates of Return | Associated Probability | |
| X | Y | Pxj=Pyj |
| 0.10 | 0.09 | 0.20 |
| 0.11 | 0.11 | 0.22 |
| 0.17 | 0.16 | 0.25 |
| 0.19 | 0.18 | 0.33 |
Find