86.3k views
4 votes
One model for the spread of a rumor is that the rate of spread is proportional to the product of the fraction y of the population who have heard the rumor and the fraction who have not heard the rumor. (a) Write a differential equation that is satisfied by y. (Use k for the constant of proportionality.)

1 Answer

5 votes

Answer:

Therefore ,
y= (Ae^(kt))/(1+Ae^(kt) )

Explanation:

The fraction of population who have heard rumor = y

The fraction of population who haven't heard rumor = 1-y

The rate of of spread (y'(t)) is proportional to the product of the fraction of population who have heard rumor the fraction of population who have heard rumor.

Therefore

y'(t) ∝ y (1-y)

⇒ y'(t) =k y (1-y) [ k = constant of proportional]


\Rightarrow (dy)/(dt)=ky(1-y)


\Rightarrow (dy)/(y(1-y))=k \ dt

Integrating both sides


\Rightarrow \int(dy)/(y(1-y))=\int k \ dt


\Rightarrow \int (dy)/(y)+\int(dy)/(1-y)=\int k \ dt
[\because (1)/(y(1-y)=(1)/(y)+(1)/(1-y) ]


\Rightarrow ln \ y- ln \ |1-y| = kt +c [ c = arbitrary constant]


\Rightarrow ln|(y)/(1-y)|=kt +c


\Rightarrow (y)/(1-y)= e^(kt+c)


\Rightarrow (y)/(1-y)= Ae^(kt) [ Here
e^c=A ]


\Rightarrow y = Ae^(kt)(1-y)


\Rightarrow y = Ae^(kt)-y Ae^(kt)


\Rightarrow y+y Ae^(kt) = Ae^(kt)


\Rightarrow y(1+Ae^(kt) )= Ae^(kt)


\Rightarrow y= (Ae^(kt))/(1+Ae^(kt) )

Therefore ,
y= (Ae^(kt))/(1+Ae^(kt) )

User PhysicsGuy
by
8.3k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories