Nasty differnation variables things

[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 don't know where to start

Can someone walk me through B please :)

 

[rock.freak667]

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

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


integrate both sides w.r.t. t

 

[thomas49th]

Can you show me how. I don't know how to intergrate this equation

Thanks :)

 

[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?

 

[thomas49th]

this is separation of variables?

 

[rock.freak667]

this is separation of variables?

Yes

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



are you able to do the left side?