y=90/(3+12e^-t) i.e. Logistic Function
Find an initial-value problem(diff. EQ in terms of y), whose solution is the y(t) above
To find an initial-value problem whose solution is given by the provided logistic function, we need to derive a differential equation involving y.
Given the logistic function y = 90 / (3 + 12e^(-t)), we can start by taking the derivative of y with respect to t:
dy/dt = d/dt [90 / (3 + 12e^(-t))]
Next, we'll need to use the quotient rule to differentiate the right-hand side of the equation:
dy/dt = [(0)(3 + 12e^(-t)) - 90(0)(-e^(-t))] / (3 + 12e^(-t))^2
Simplifying this expression yields:
dy/dt = (0 - 0) / (3 + 12e^(-t))^2
dy/dt = 0
Hence, the differential equation associated with the given logistic function is:
dy/dt = 0
Now, let's proceed with finding the initial-value problem by providing an initial condition, such as y(0) = 10:
Therefore, the initial-value problem is:
dy/dt = 0, with initial condition y(0) = 10