# ODE bacterial growth problem

1. Feb 8, 2010

### S.N.

1. The problem statement, all variables and given/known data

A cell of some bacteria divides into two cells every 40 minutes. The initial population is 2 bacteria.

a)Find the size of the population after t hours
b) Find the size of the population after 6 hours.
c) When will the population reach 12?

2. Relevant equations

None given.

3. The attempt at a solution

taking t = time in hours, y(t) = population,

we say that

dy/dt = (3/2)y --- the 3/2 converts from hours to the 40 min growth period
dy/y = (3/2)dt
integrate....
ln(y) = (3/2)t + C --- C is arbitrary constant
y = [e^( (3/2)t )] * L -- L is arbitrary constant analogous to e^C.

we know that y(0) = 2, so to solve initial value problem,

2 = 1*L

therefore L = 2

and the solution for part a) should be y(t) = 2 e^[(3/2)t]

but this solution is unfortunately incorrect and I don't see how to fix it. Of course it holds for y(0) as it should, but it falls apart when I check other values. a friend mentioned something about "adding or subtracting a 1" somewhere but I don't see where I would do that, and even if I did I wouldn't understand the justification.

Any help is appreciated, thanks!

2. Feb 8, 2010

### Kizaru

A suggestion: convert to hours at the very end.