Separation of Variables and the Logistic Equation (6.3) January 30th, 2017

Separation of Variables *Recall that a differential equation can only be solved through separation of variables by multiplication or division (no addition or subtraction).

Ex. 1: Find the general solution of the differential equation. b.

Ex. 2: Find the particular solution that satisfies the initial condition. b.

Orthogonal Trajectories *Two families of curves are mutually orthogonal, and each curve in one of the families is called an orthogonal trajectory of the other family if it forms a right angle with one of the curves in the other family.

Ex. 3: Find the orthogonal trajectories of the family of curves given by .

The Logistic Differential Equation *Unlike a basic exponential growth model, most populations have some upper limit L past which growth cannot occur. These can be modeled by the logistic differential equation, , where k and L are positive constants and L is called the carrying capacity, which is the maximum population y(t) that can be supported or sustained as t increases. If , the population increases, whereas if , the population decreases.

As ,

Ex. 4: The population of Nowheresville is 20,000 people Ex. 4: The population of Nowheresville is 20,000 people. After 4 years, the population is 22,500 people. It is believed that Nowheresville can only sustain 100,000 people. Write a logistic differential equation that models the population of Nowheresville. Find a general solution to the logistic differential equation that can be used for all logistic models. Find the logistic model for Nowheresville. Use the model to estimate the population after 20 years. Find the limit of the model as