(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

One model for the spread of 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.

i) Write a differential equation that is satisfied by y

ii) Solve the differential equation.

2. Relevant equations

3. The attempt at a solution

i) dy/dt = ky(1-y)

ii) I was having trouble solving this equation

When I took the integral I got

ln| (y^2)/(y - y^2) | + c = kt

I checked that ln| (y^2)/(y - y^2) | + c was the correct integral of 1/(y-y^2) with wolfram alpha and it was

I than began to solve for y and got a quadratic equation but than stopped solving because I checked the answer key...

"Logistic Eqn dp/dt = kp(1-p/k) so k=1, P=y

Now the solution to (i) is y(t) = k/(1 + A e^(-kt))

y(t) = 1/(1+Ae^(-kt))

At t = 0 y_0 = 1/(1+A) => 1+ A = 1/y_0

A = 1/y_0 - 1

y = 1/(1+(1/y_0 - 1)e^(-kt) ) = y_0/(y_0 + (1-y_0)e^(-kt))"

I don't understand what this equation is y(t) = k/(1 + A e^(-kt)) and have never seen it before and don't think that if I solved the equation ln| (y^2)/(y - y^2) | + c = kt I would of gotten this... I don't understand... also y_0 is y sub zero or y at time equals zero...

thanks for any help!

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# Homework Help: Calculus 2 - Introduction to Differential Equations

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