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EI
Numerade Educator

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Problem 18 Easy Difficulty

Let $ c $ be a positive number. A differential equation of the form
$ \frac {dy}{dt} = ky^{1+c} $
where $ k $ is a positive constant, is called a doomsday equation because the exponent in the expression $ ky^{1+c} $ is larger than the exponent 1 for natural growth.
(a) Determine the solution that satisfies the initial condition $ y(0) = y_0. $
(b) Show that there is a finite time $ t = T $ (doomsday) such that lim $ _{t \to T} - y(t) = \infty. $
(c) An especially prolific breed of rabbits has the growth term $ ky^{1.01}. $ If 2 such rabbits breed initially and the warren has 16 rabbits after three months, then when is doomsday?

Answer

a) $y(t)=\left(y_{0}^{-c}-c k t\right)^{-\frac{1}{c}}$
b) As $t \rightarrow T^{-}, c k T < y_{0}^{-c},$ so you have a very small positive number to a negative exponent, which results in a large positive number. Therefore the expression goes to $+\infty$ as $t \rightarrow T^{-}$ .
c) 145.8 months

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Video Transcript

Okay in this question we're going to be looking at the doomsday equation. So the first part as to determine the solution that satisfies the initial condition. So can dy DT equals K. Y. The one plus C. In order to solve this, we're going to do a separation of variables. So we're going to bring the white to the other side and then the D. T. To the other side. And we're going to integrate this. So I'm going to rewrite this as why to the negative one minus C. Dy equals the integral of K. D. T. And when you solve that you get negative Y to the negative C over. See equals Katie plus a constant. Then I'm going to call each and then you can actually just solve for why why is going to be equal to this constant H minus C. Okay, t all raise the negative one over. See, okay, now we want to and put the initial condition Y of zero equals you plug in zero here. So this term becomes zero and this is going to be equal to why not? So this means that h equals why not raise the negative one, oversee, sorry, not one oversee. Just to the sea power. It's a negative C. So that means Y. Of T equals why not? The negative c minus K. C. T raised a negative one over. See that is the first part. The second part asked show that there is a finite time such that the limit as T approaches the time equals infinity. So to break this down the Y of T. Is because you have a negative exposing here, you could write this as Y. Of T equals one divided by why not the negative c minus K. C. K. T. To the one oversee. So if the denominator why not the negative c minus C. K. T. Race of the one oversee. But that doesn't really matter. That equals zero. Then you're going to have a one divided by zero case and that's a dis continuity and it will explode to infinity. So that's what's meant by the doomsday scenario. Is that in a certain amount of time, your functions can explode to infinity. Now we can actually solve for what this value is going to be by solving this equation. So we have and we want to solve for the time. So we have why not? The negative C equals C. K. T. So that means that the big T that we're looking for is why not negative C divided by see Okay. All right, the last part is a specific example. So we have the growth rate of dy DT equals ky to the 1.1. So from here we can immediately tell that C. Is 0.1. And we're also given that why not equals two. So we plug those into the equation that we have Y. Of T equals to raise the negative 0.1 minus 0.1 times K. Times three. Um Well not I'll just write this out as T. Right now. Raised to the negative one divided by 0.1. And were given that Y. Of three equals 16. So if we plug in three for T we get negative 0.3. Okay raised to the negative one divided played by 10.1. Okay. And now we can solve for K. And if you do that you get Kay is approximately 2.68. So now we can use that to find our doomsday T. Equals why not? The negative C. Divided by C. K. We have all the values we need to to the negative 0.1 divided by 0.1 at times 0.68. And when you do that you get approximately 146. And the time that we're looking at is months because I plugged in Y. Of three. That was three months. So there is doomsday.