This is not separable. However, if I write it as follows:
This equation has the form of a Bernoulli's Equation (I cover this in my Differential Equations Tutorial):
(general form of Bernoulli Equation)
If we make a substitution the Bernoulli Eqn will become a linear equation we know how to solve.
We also need a substitution for .
We can get this by applying a definition of the chain rule: .
Substituting this into the DE, we get:
Multiply both sides by to get a linear DE:
Apply Integrating factor:
It should be known by now: when we multiply through by the int. factor, we get on the left side of the equation the derivative of
Thus, our DE becomes:
Solving the DE, we get:
We don't want v. We want y. So sub back in
When , we have:
I believe in this case and
Since y(2)=50, we see that
My calculator tells me that
Now find t when y(t)=80
So at 1 PM, 80 students will have heard the rumor.
(I hope my math was right! It does make sense!)
Hope this answered your question!