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# Differential Equations/Bacteria Growth model

Question
The Original Problem:
The population of bacteria in a culture grows at a rate proportional to the number of bacteria present at time t. After 3 hours it is observed that 400 bacteria are present. After 10 hours 2000 bacteria are present. What was the initial number of bacteria.

I'm struggling with the whole concept of DE. In this case, I'm not sure where to start with this application. When given the inital value, I think I know how to find C. But this is confusing to me.

P(t) is the population of the bacteria at time t.
So P(3)=400 and P(10)=2000

If P(t)=c*e^(rate*t), and t=0, then finding C makes sense to me. But if I don't know P(0), then how can I find C when I end up with two variables? Is this like solving a system of equations in Algebra because I have two populations at different times? I guess I just need to know how to start this.

Thanks.

P=c*e^(r t)
t=3 , P = 400:  400 = c*e^(3r)
t=10, P = 2000: 2000 = c*e^(10r)
Solve for r and c from these equations.
2000/400 = e^(10r)/e^(3r)
5 = e^(7r)
e^r = 5^(1/7)
Find c:
c*e^(3r) = 400:  c = 400*e^(-3r) = 400*5^(-3/7)
Solution:
P = 400 * 5^((t-3)/7)

Differential Equations

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#### Dr. Nyayapati Swami

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