Nasty differnation variables things

1. Jun 11, 2008

thomas49th

A population grows in such a way that the rate of change of the population P at time t in days is proportional to P.

a) Write down a dfferential equation relating P and t

b) Show, by solving this equation, that the general solution of this equation may be written as $$P = Ak^{t}$$, where A and k are positive constants.

a) is easy:

dP/dt = kP

b) I dont know where to start

Can someone walk me through B please :)

2. Jun 11, 2008

rock.freak667

$$\frac{dP}{dt} = kP$$

$$\Rightarrow \frac{1}{P}\frac{dP}{dt}=k$$

integrate both sides w.r.t. t

3. Jun 11, 2008

thomas49th

Can you show me how. I dont know how to intergrate this equation

Thanks :)

4. Jun 11, 2008

thomas49th

hang on this is one of those stupid natural log ones

i know if y = a ^ x then dy/dx = a^x ln a

but how does that help?

5. Jun 11, 2008

thomas49th

this is seperation of variables?

6. Jun 11, 2008

rock.freak667

Yes

$$\int \frac{1}{P}dP= \int k dt$$

are you able to do the left side?