5 Steps to a 5 AP Calculus BC 2019

328 STEP 4. Review the Knowledge You Need to Score High

Step 5: Substitute initial condition (0, 3) and obtaink=3. Thus, you havey=

3

e

x 22.

3

2

1

–2 0–1 1 2

y

x

decr.

Figure 13.5-7

13.6 Logistic Differential Equations

Main Concepts:Logistic Growth

Often a population may grow exponentially at first, but eventually slows as it nears a limit,

called the carrying capacity. This pattern is called logistic growth, and is represented by the

differential equation

dP

dt

=kP

(

1 −

P

K

)

, in whichPis the population,Kis the carrying

capacity, andkis the proportional constant. The differential equation is separable so

dP

dt

=

kP

(

1 −

P

K

)

⇒

dP

dt

=

kP(K−P)

K

⇒

KdP

P(K−P)

=kdt. This equation can be integrated

using a partial fraction decomposition.

∫

KdP

P(K−P)

=

∫

kdt

∫(

1

P

+

− 1

K−P

)

dP=

∫

kdt

ln|P|−ln|K−P|=kt+C 1

ln

∣∣

∣∣ P

K−P

∣∣

∣∣=kt+C 1