ODE Rumour Spread

Question

A certain person starts a rumour in a small town. The number of people who have heard the rumour, $R(t)$, is given by $$ \frac{\text d R}{\text d t} = KR \left(1300 - R\right) $$ where $K$ is a positive constant, and 1300 is the number of residents in that small town.

By regarding this equation as a Bernoulli equation, find $R(t)$.

This is my current working, I have no idea how the answer got $\frac1R=\frac1{1300}+Ce^{-1300Kt}$

Where did my t go to?

Followed by solving the equation to get $\frac1R=\frac1{1300}+\frac{1299}{1300}e^{-1300Kt}$

Answer 1

Simply write your equation as \begin{align} \frac{dR}{kR(1300-R)} = dt \end{align} and integrate! [I figure out you did typo on your notes!]

And then managing $t$ then you will get the answer.

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About @deviljones comment, try $x=\frac{1}{R}$[since $R=R(t)$, $x=x(t)$], then \begin{align} -\frac{dx}{1300kx-k} = dt \end{align} after integration you have \begin{align} ln(k-1300kx) = -1300kt + C \end{align} Then expressing in terms of $x=\frac{1}{R}$, you get what you want.

Answer 2

Hint

It would be easier to solve $$\frac 1{t'}=k R(1300-R)$$ Reverse it, use partial fraction decomposition to get $t(R)$ and then $R(t)$.