The following logistic model can be used to estimate the population of Antarctic

Question

The following logistic model can be used to estimate the population of Antarctic fin whales (in tens of thousands) with the assumption of harvesting (removal of population) at a rate proportional to the size of the population. Verify that the indicated expression is an implicit solution of the given differential equation. Assume an appropriate interval I of definition for the solution.

*Use Implicit Differentiation to solve.

2. The following logistic model can be used to estimate the population of Antarctic fin whales (in tens of thousands) with the assumption of harvesting (removal of population) at a rate proportional to the size of the population. Verify that the indicated expression is an implicit solution of the given differential equation. Assume an appropriate interval I of definition for the solution dt 25 240 20-y 25

Explanation / Answer

ln(y / 20-y ) =t/25

=> ln(y) - ln(20-y) = t/25

differentiate both sides

=> d(ln(y) - ln(20-y)) = d(t/25)

=>(1/y)*dy - (1/20-y)* -1 * dy = 1/25 * dt

=> dy(1/y + 1/20-y) = dt/25

=> dy(20-y+y / y(20-y)) = dt/25 (Next step : taking lcm of y and 20-y and solve)

=>20 dy / y (20-y) = dt/25 (Next step : send dt to left side and every element except dy to right side )

=>dy/dt = (1/25)*y*(20-y)/20 (Next step : divide last bracket by 20)

=> dy/dt = (1/25)*y*(1 - y/20) (Next step : take 2 common )

=> dy/dt = (1/25)*y*2*(1/2 - y/40) (Next step : rearranging)

=> dy/dt = (2/25) y (1/2- y/40) Proved

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