# ODE Population problem

I know that the rate of change with time of a population is proportional to the square root of t. T=0 is y = 100. Population increases at rate of 20 per month.

I started out by trying to do dy/dt = p^.5. I am used to the population problems where I use y=Ce^(rt) but am having trouble making the jump to this kind. How do you account for the increase of 20/month?

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The population always increases at a rate of 20 per month, or is that an initial value?

Yes, always increases at rate of 20/month and initial population is 100.

Than the question makes no sense to me. How can the rate of increase in the population be of 20/month and proportional to the square root of t at the same time?

You and me both then. Here is the exact question:
The time rate of change of a rabbit population P is proportional to the square root of P. At time t = 0 (months) the population numbers 100 rabbits and is increasing at the rate of 20 rabbits per month. How many after one year? The answer I guess is 484.

How do you set up logistics equations?

So 20/month is an initial value.

The rate of change of P, i.e. it's derivative with respect to P is proportional to the square root of P:

$$\frac{dP(t)}{dt} = \alpha P^{1/2}$$

where alpha is a constant with respect to. You also know that

$$\frac{dP(t)}{dt} \bigg|_{t=0}} = 20 \ rabbits/month$$

and that

$$P(0) = 100 \ rabbits$$

Which is all the info you need to determine P(t)

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I appreciate what you wrote but I'm still unsure on where to start. I see that I was wrong to have a p^.5 on the right. Should I go ahead and start separating variables and then moving forward?

I appreciate what you wrote but I'm still unsure on where to start. I see that I was wrong to have a p^.5 on the right
OOOPSSS, my mistake. I corrected it in the previous post.

Should I go ahead and start separating variables and then moving forward?

Of course you should. Even in the case where you shouldn't, equations don't bite you know.

But how do you incorporate the dp/dt = 20 and P(0) = 100? I've done the sep of variables, no problem there. Sitting on 2p^.5 = (alpha)t + C.

Made another attempt at this, can you tell me where I'm going wrong?
I came up with this equation after sep of variables: 2p^.5 = Bt + C.
1) With t = 0 and initial population of 100, I came up with a C value of 100.
2) With t = 1, population = 120, so I came up with a "B" value of 120.
3) But then when I do time = 12, I don't get anywhere near the 484 rabbits that I should have.

Matterwave