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Problem 52 Hard Difficulty

A model for tumor growth is given by the Gompertz equation
$ \frac {dV}{dt} = a (\ln b - \ln V) V $
where $ a $ and $ b $ are positive constant and $ V $ is the volume of the tumor measured in $ mm^3. $
(a) Find a family of solution for tumor volume as a function of time.
(b) Find the solution that has an initial tumor volume of $ V(0) = 1 mm^3. $

Answer

a) $V=b \cdot e^{-e^{-c^{C-a t}}}$
b) $V=b^{1-e^{-a t}}$

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Video Transcript

Okay, So this problem gives us, uh, the differential equation that we can split comes We have DVD t equals eight times natural. I could be minus natural like a V at times. Be Yes, we can ah, take apart this expression. So what I'm gonna do is instead, right? This is DVD T is equal to the negative. He, uh, times natural law, Gov Afghanistan Shal I could be And then times a Okay, so now we're gonna pull the whole first part of this equation. It's our last night and multiply both sides by DT. It's gonna have DVD over for dividing both sides. By that will have v times natural like of the minus natural. Your be equals negative a d T. Okay, so what we can do here is say, since natural longer be is just a constant Well, we can set. Are you as natural, longer being? Yeah. This is a little bit difficult because you ve times both of them. So, um, if are you is equal to I'm not trying to be. Andi would be equal to one over acts. Okay, so then we can write this as, say, one of Rex Devious sorry we can We can change our last side to be the integral because we're integrating both sides of do you? I don't find it by you. Minus the natural Could be okay. Was it a difficult in a girl? Just simplify as we go. Um, the case of then if Ah, and what do we do next? Was that, uh let's just do an ass. So Stewart s is equal to you. Might s natural. Longer be. And yes, just be dio this this becomes integral of one over S t s king and that's if we integrate that that's going to give us the natural log top us and then s was equal to u minus The natural log of B Then you was the natural longer v Kim had to about Okay, so we have natural log of the natural look of V Chronis the natural Huckabee. Okay. They wouldn't normally have close a constant, but we can combine are constants. Are there other side just gonna be negative A t and plus c who? Okay, still not done because we need to. We need the family of equations that this would work for, So we need to solve for V. Just simplifying our left side will have natural log of natural log of the Overbey. We'll be more comfortable with the out group this because negative eight times t plus c I came and raised both sides today. Ah, to the power. So we're gonna get natural log of the ever being oh equals yeah, to the C minus 80. Just a friend of Mr Negative somewhere, so I mean, double check real quick. Okay, Pull the negative out here. I left the negative on the right side. I'm setting you is natural. Give the okay. Eso have e to the C minus 80. Then I'm a raise each side to the power again. So I'm gonna have V over being his eagle to e to the to the C minus 80. Power and multiplying both sides about be I have V A t is equal to be times e to the to the C minus 80. Okay, now, from here, uh, the second parts. This is our family of functions. This thing of art says that when t is equal to zero, V is equal to one. So we plug in zero, we'll have one is equal to be, uh, times e to the to the sea power, because minus 80 is just gonna become zero. Um, that means we can righty to the e to the C power, as one ever be. I'm so we can kind of help solve this, Kim. So now we can say that RV of tea going on minus 80. Back in, we can say it be times one Overbey to the E to the negative 80. Power eyes are function, and we might be able to simplify this further. So it's called his B to the first times be to the negative. First raised to the E to the negative 80 power. And so this function should be be, uh, to the one minus e to the negative 80 power? No, but it should still work. As if t zero that should give us a zero. Um, negative one. And you melt power to power. You multiply. So that will be our final equation. Okay, thank you very much.