Question

True or False The graph of any solution to the differential equation \(d P / d t=k P(100-P)\) has asymptotes \(y=0\) and \(y=100 .\) Justify your answer.

Short answer

True, the graph of any solution to the differential equation has asymptotes y=0 and y=100.

Step-by-step solution

Understanding the Differential Equation and its Solutions

The given differential equation is \[ \frac{dP}{dt}=kP(100 - P) \]. This is a separable differential equation and one of the solutions is a logistic function, often used to represent population growth. The logistic function \( P(t) \) here represents a population, and \( 100 \) is the carrying capacity of the environment.

Analyzing the Asymptotic Behavior

An asymptote is a line that a curve approaches, as it heads towards infinity. For values of \( t \) very large in (positive or negative) magnitude, a solution to the differential equation is said to be an asymptote. For the solution to the equation, analyze the possible values for the function as \( t \) approaches infinity. Start with the special cases: If \( P(t) = 0 \), then \( \frac{dP}{dt} = 0 \), so constant \( P = 0 \) is a solution. Similarly, if \( P(t) = 100 \), then \( \frac{dP}{dt} = 0 \), so constant \( P = 100 \) is a solution. Both of these are equilibrium solutions to the differential equation.

Concluding the Asymptotes

Because the logistic function describes a population that grows from an initial small population to its maximum capacity, it starts close to 0 when \( t \) is very small and eventually approaches 100 as \( t \) becomes large. Therefore, the graph of logistic function \( P(t) \) has horizontal asymptotes, \( y = 0 \) and \( y = 100 \), which means the statement in the problem is true.