Suppose that the population of cats in a has a rate of growth proportional to the population itself. Write down a differential equation for the population $P (t)$ at time $t$ ?
What is the differential equation expression for this question?
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0Yes, but I am only having trouble getting the expression and not the solving part. – 2011-03-14
3 Answers
Well, we know that $k*P(t)=dP/dt$. This simply states that $P(t)$ is proportional to $dP/dt$.
You may be able to stop here, based on the wording of your question. It simply asks for a differential equation involving P(t) and t, which this classifies as. You can move on if you have to solve for k at some point.
First, we rearrange this to be $dP/P = kdt$.
If we integrate both sides, $\int dP/P = \int kdt$
This leaves us with $ln(P) = kt + C$
Then, we do the opposite operation of the natural log (putting everything to the power of e) and we are left with: $P = e^{kt+c}$
Then you solve for the initial y and the final form of the equation is:
$P=P_{0}e^{kt}$
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0I originally stated you could stop at the second step; you may actually be able to just use the function in the first sentence for your answer! – 2011-03-14
Hint: how would you express the rate of growth in terms of $P(t)$?
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1@Kartik: $\frac{dP}{dt} = rP(1-\frac{P}{K})$ is not the answer to "a rate of growth proportional to the population itself"; it is instead a variant of the [logistic function](http://en.wikipedia.org/wiki/Logistic_function#In_ecology:_modeling_population_growth) with $K$ being the *carrying capacity* or maximum possible number of cats. But your question makes no mention of any maximum. – 2011-03-14
To the OP: the equation you stated in a comment to Ross is crucial (and in contradiction with your post, I should say).
So I understand that the question is to solve P'(t)=rP(t)(1-P(t)/K), presumably with an initial condition $P(0)$ in $(0,K)$ (and if this is not what you ask for, please say so and I shall delete this answer).
Hint: The differential equation P'(t)=rP(t)(1-P(t)/K) can be rewritten as f(P(t))P'(t)=r for a function $f$ you should be able to write down. Let $g$ be any differentiable function and $G(t)=g(P(t))$. Compute G'(t). Choose $g$ such that G'(t)=P'(t)f(P(t)). The equation G'(t)=r is equivalent to $G(t)=rt+c$ for a given constant $c$. Use all this to solve the original equation.