Separable D.E., Partial Fractions

[EtherealMonkey]

Homework Statement

$$\frac{dP}{dt}=P-P^{2}$$

It seems that Partial Fractions should be used to solve this D.E., but I cannot find an example to go by.

I even tried to rewrite the equation as:

$$\frac{d}{dx}Y\left(x\right)=Y\left(x\right)-Y\left(x\right)^{2}$$

But, that isn't helping me either...

Anyone?

 

[EtherealMonkey]

Please?

I know this may seem so obvious. And, in fact a great deal of my trouble in D.E. so far can be summarized by my "not seeing the forest for the trees".

So, any hint would probably get me going...

 

[EtherealMonkey]

This is what I have done so far...

$$P\left(t\right)\frac{d}{dt}=P\left(t\right)-P\left(t\right)^{2}$$

$$P\left(t\right)\frac{d}{dt}=P\left(t\right)\left(1-P\left(t\right)\right)$$

$$\frac{1}{P\left(t\right)}dP=\left(1-P\left(t\right)\right)dt$$

$$\int \frac{1}{P\left(t\right)}dP= \int dt - \int P\left(t\right) dt$$

$$\ln \left(P\left(t\right)\right) = t - \int P\left(t\right) dt$$

 

[Mark44]

This is what I have done so far...

$$P\left(t\right)\frac{d}{dt}=P\left(t\right)-P\left(t\right)^{2}$$

$$P\left(t\right)\frac{d}{dt}=P\left(t\right)\left(1-P\left(t\right)\right)$$

$$\frac{1}{P\left(t\right)}dP=\left(1-P\left(t\right)\right)dt$$

$$\int \frac{1}{P\left(t\right)}dP= \int dt - \int P\left(t\right) dt$$

$$\ln \left(P\left(t\right)\right) = t - \int P\left(t\right) dt$$

I see you were able to get the 'y' in your user name!

One comment: you're still using the derivative operator incorrectly: d/dt requires something to the right of it, since it means to take the derivative with respect to t of something.

Now, let's look at your DE.
$$\frac{dP}{dt}~=~P~-P^2$$
After separation, you get this.
$$\frac{dP}{P - P^2}~=~dt$$

Now integrate both sides. You will want to use partial fractions for the integral on the left side, and should get two terms involving ln|P| and ln|1 - P|. Don't forget you'll need the constant of integration.

Is that enough to get you started?

 

[EtherealMonkey]

Is that enough to get you started?

Yes, thank you very much!

Sorry for the late response, I just got on campus.

Now, got to go take a test, wish me much success!

Thanks again Mark!