The population of bacteria in a culture grows at a rate proportional to the number of bacteria present at time t [hrs.]. After 3 hours, 400 bacteria are present. After 10 hours, 2000 are present.

What was the initial number of bacteria?

Question

The population of bacteria in a culture grows at a rate proportional to the number of bacteria present at time t [hrs.]. After 3 hours, 400 bacteria are present. After 10 hours, 2000 are present. 

What was the initial number of bacteria?

danjs · Answer

I'll give a hint - It is a linear first order ODE. $$\frac{ dP }{ dt } \alpha  (P)$$
so
$$\frac{ dP }{ dt } = kP$$

misty1212 · Answer

i will give you a bigger hint
forget ode and fancy pants stuff 
after 7 hours it increases by a factor of $2000\div 400=5$
use 
$$P=400\times 5^{\frac{t}{7}}$$ 
you want to know what it was 3 hours earlier, plug in $t=-3$

misty1212 · Answer

http://www.wolframalpha.com/input/?i=400*+5^%28-3%2F7%29

danjs · Answer

Right, I am looking for the solution method to the general form of:  $$\frac{ dy }{ dx } + p(x)*y = q(x)$$
using an inegrating factor
$$u(x) = e ^{\int\limits p(x)*dx}$$

ganeshie8 · Answer

$$\large  5^{t/7} = e^{(\ln 5/7)t}$$

ganeshie8 · Answer

why do you even need to solve a linear ODE ? your equation is separable right ?

ganeshie8 · Answer

$$\frac{ dP }{ dt } = kP$$

$$\int\frac{ dP }{ P } = \int k~dt$$

ganeshie8 · Answer

solving gives you the same eqn used by misty

misty1212 · Answer

why do you need to solve anything? you are given all the numbers you need in the question

ganeshie8 · Answer

@misty1212  the op is trying to derive that growth formula using calc it seems

danjs · Answer

Right, it is just a easy beginner problem as an intro to solving 1st order linear equations constant coefficients.

danjs · Answer

I just put it up for people to mess with.

misty1212 · Answer

lol

ganeshie8 · Answer

its not a general linear ODE
as you can see it is separable.. i think you need to add some gangsters that steal money from you every month or so to make it more general..

danjs · Answer

yes it is also separable i know..lol fine ill post one that is not separable, and needs to be solved with an integrating factor.

ganeshie8 · Answer

$$\frac{ dP }{ dt } = kP - 100t$$


100$ goes out eery time unit^
now it is no longer separable and you will have to use IF

ganeshie8 · Answer

*every

danjs · Answer

yeh, i was just making it easier. ill post a new one

ganeshie8 · Answer

I see.. y' = ky is linear but it is also separable and autonomous so we we solve it by separating variables
adding some time factor on right hand side makes it impossible for separating variables

mathmath333 · Answer

what is the answer

mathmath333 · Answer

is it 252

misty1212 · Answer

no 200

danjs · Answer

Just

$$P(t) = P _{0}e ^{kt}$$

Po is about 201 bacteria

danjs · Answer

$$k = \frac{ 1 }{ 7 }\ln(5)$$