Question

Consider the logistic equation ˙=(1−) y ˙ = y ( 1 − y ) (a) Find the solution satisfying 1(0)=14 y 1 ( 0 ) = 14 and 2(0)=−3 y 2 ( 0 ) = − 3 .

Answer 1

y(t) = 1 / (1 + (-4e^(-t))

Explanation:

The logistic equation is a nonlinear differential equation that describes how a population grows. The solution for the logistic equation is given by y(t) = 1 / (1 + (Ce^(-t)) where C is the constant of integration, which can be determined by the initial conditions.

For the first initial condition, 1(0)=14, we can substitute the initial condition into the solution to get:

y(0) = 1 / (1 + (Ce^(0)) = 1 / (1 + C) = 14

Solving for C, we get C = 13.

So the solution for the first initial condition is:

y(t) = 1 / (1 + (13e^(-t))

For the second initial condition, 2(0)=-3, we can substitute the initial condition into the solution to get:

y(0) = 1 / (1 + (Ce^(0)) = 1 / (1 + C) = -3

Solving for C, we get C = -4.

So the solution for the second initial condition is:

y(t) = 1 / (1 + (-4e^(-t))

Answer 2

The logistic equation is a differential equation that describes the growth of a population over time. The solution of this equation can be found using separation of variables or other methods of solving differential equations. The general solution for the logistic equation is given by:

y(t) = y_0 * e^(rt) / (1 + (y_0 - 1)e^(rt))

Where y_0 is the initial population size, r is the growth rate, and t is time.

For the specific case given in the question, we have the initial conditions y1(0)=14 and y2(0)=-3. We can use these to find the specific solution for each case.

1(0)=14 => y1(t) = 14e^(rt) / (1 + (14 - 1)e^(rt))

2(0)=−3 => y2(t) = -3e^(rt) / (1 + (-3 - 1)e^(rt))

It is important to note that we can not find the exact value of r without additional information.