beginner differential equation

[burt]

Suppose a population at time t, P(t), satisfies 10P'(t)=P(t). If P(0)=2, what is P(20)?

I was given this question. I am nearing the end of calculus 1 and my book sticks this in kind of as an aside. I know that P=Ce^{kt} and I also know that \(\displaystyle \frac{dP}{dt}=kP\). However, I cannot make heads or tails out of this problem - I have no idea what to do? What steps should I take in order to solve the problem?

 

[MarkFL]

Given:

[MATH]\frac{dP}{dt}=kP[/MATH]
Usually, the first method taught for solving such an ODE is to separate variables and integrate:

[MATH]\frac{1}{P}\,dP=k\,dt[/MATH]
Can you proceed?

 

[burt]

Given:

[MATH]\frac{dP}{dt}=kP[/MATH]
Usually, the first method taught for solving such an ODE is to separate variables and integrate:

[MATH]\frac{1}{P}\,dP=k\,dt[/MATH]
Can you proceed?

Not quite yet.

 

[MarkFL]

Not quite yet.

How about now?

[MATH]\int \frac{1}{P}\,dP=k\int \,dt[/MATH]

 

[HallsofIvy]

You say you know that "\(\displaystyle P(t)= Ce^{kt}\)". How do you know that? Is that given? In any case, from that, \(\displaystyle \frac{dP}{t}= kCe^{kt}= kP(t)\). Do you recognize that, in this problem, k= 1/10?

 

[Deleted member 4993]

Going more basic:

Can you differentiate

ln(P) with respect to P.

Or calculate:

\(\displaystyle \displaystyle{\dfrac{d}{dP}[ln(P)]}\) = ?