Kenneth Kuttler

550 CHAPTER 29. FIRST ORDER SCALAR ODE

The next example has to do with population models. It was mentioned earlier. Theidea is that if there were infinite resources, population growth would satisfy the differentialequation

dydt

= ky

where k is a constant. However, resources are not infinite and so k should be modified tobe consistent with this. Instead of k, one writes r

(1− y

)which will cause the population

growth to decrease as soon as y exceeds K. Of course the problem with this is that we arenot sure whether K itself is dependent on other factors not included in the model.

Example 29.3.7 The equation

dydt

= r(

1− yK

)y, r,K > 0

is called the logistic equation. It models population growth. You see that the right side isequal to 0 at the two values y = K and y = 0.

This is a separable equation. Thus

dy(1− y

)y= rdt

Now you do∫

to both sides. This requires partial fractions on the left.

1(1− y

)y=

1K− y

+1y

Therefore,ln(y)− ln(K− y) = rt +C

if 0 < y < K. If y > K, you get

ln(y)− ln(y−K) = rt +C

Therefore, the integral curves are of the form

ln(

yK− y

)= rt +C

so changing the name of the constant C, it follows that for y < K, the integral curves aredescribed by the following function.

y = KCert

Cert +1, C > 0

In case y > K, these curves are described by

y = KCert

Cert −1, C > 0

What follows is a picture of the slope field along with some of these integral curves in caser = 1 and K = 10.

550 CHAPTER 29. FIRST ORDER SCALAR ODEThe next example has to do with population models. It was mentioned earlier. Theidea is that if there were infinite resources, population growth would satisfy the differentialequationdydtwhere k is a constant. However, resources are not infinite and so k should be modified tobe consistent with this. Instead of k, one writes r (1 — z) which will cause the populationgrowth to decrease as soon as y exceeds K. Of course the problem with this is that we arenot sure whether K itself is dependent on other factors not included in the model.Example 29.3.7 The equationdy _y= 1-=) KGi a1 (1g) HK > 0is called the logistic equation. It models population growth. You see that the right side isequal to 0 at the two values y = K and y = 0.This is a separable equation. Thusdy(l1-x)yNow you do f to both sides. This requires partial fractions on the left.= rdt1 11+=(1-z)y K-y yTherefore,In(y) —In(K —y) =rt+Cif0<y<K.Ify>K, you getIn(y) —In(y—K) =rt+CTherefore, the integral curves are of the formyIn ( —— ]) =7t+Cn( 2) rt+so changing the name of the constant C, it follows that for y < K, the integral curves aredescribed by the following function.Ce”= K——_,,C>0oS" Cert]In case y > K, these curves are described byCe™= K———_,,C > 0y Ce™—1What follows is a picture of the slope field along with some of these integral curves in caser=landk =10.