Differential Equation Homework: Solving 1000(dp/dt) = p(100-p)

[Illusionist]

Homework Statement

Hi, I've been trying to solve the differential equation 1000(dp/dt) = p(100-p), but have had no luck so far.

Homework Equations

I think this requires using the change of variables formula.

The Attempt at a Solution

Basically I've tried putting everything involving the dependent variable p, which lead me to [1000/(100p-p^2)]*(dp/dt)=1. I then tried to differentiate both sides but this is where I seem to become stuck. I know the answer is p= 200/(2-e^-0.1t) but can't get to it because of the integration.

I'm also having similar problems with the question dP/dt=P(1-0.01P)-h. Thanks in advance or any help or tips.

 

[Benny]

Just try separation of variables.

$$1000\frac{{dp}}{{dt}} = p\left( {100 - p} \right) \Rightarrow \int {\frac{{dp}}{{p\left( {100 - p} \right)}} = \frac{1}{{1000}}\int {dt} }$$

For the second one, set h = 0 and solve the ODE. Then solve dp/dt = -h and add the two solutions.

 

[Illusionist]

Yeah I got that far but I am having trouble differentiating the RHS of the equation.

 

[Mathgician]

do you remember integrating partial fractions? If not you should look it up. The right side looks pretty straight forward to me. Integrating it should be one step process. I don't know why you say RHS is a problem.

 

[Benny]

The next step following from what was in my last post would be to integrate both sides. The integral of 1 with respect to t is just t and the integral of (p(100-p))^-1 is fairly easy once you split it into partial fractions.

 

[Illusionist]

Of course, don't know what I was thinking. Sorry. Thanks a lot.