Answer to Question #121090 in Differential Equations for Joseph Se

Question

1. A given field mouse population satisfies the differential equation dp/dt = 0.5p-330 where p is the number of mice and t is the time in months.

(a) Find the time at which the population becomes extinct if p(0)=620. Round your answer to two decimal places. The absolute tolerance +/- 0.01.

tf = ____________ (months)

(b) Find the initial population p0 if the is to become extinct in 1 year. Round your answer to the nearest integer. The absolute tolerance +/- 0.01.

p0= ________ (mice)

Answer 1

$\frac{dp}{dt} = 0.5p - 330$

(i) Population at any time is given by

$\int {dp} = \int (0.5p-330 )dt$

$\int \frac{dp}{ (0.5p-330 )} = \int dt$

solving integral, we get

$2log( 0.5p-330 ) = t + c$ , where c is integral constant. . . . . . . . . . (a)

for finding the time for extinction of the population,

$2[log (0.5p-330 )]_{620}^{0} = t|_0^t$

it will give,

$t = 5.61 months$ (approx)

(ii) Using equation (a) to find the initial population so that it become extinct in one year

Equation will be

$2log[0.5p-330 ]_p^0 = t|_0^{12}$

solving this equation for p, we get $p=658.36 \approx$ $658.$