for-the-following-problems-set-up-and-solve-the-differential-equations-a-drug-is-administered-to-a-p

Question

For the following problems, set up and solve the differential equations.
A drug is administered to a patient every 24 hours and is cleared at a rate proportional to the amount of drug left in the body, with proportionality constant $0.2 .$ If the patient needs a baseline level of 5 $\mathrm{mg}$ to be in the bloodstream at all times, how large should the dose be?

Cyrielle Lorio · Accepted Answer

Okay, So what we have here is a problem involving a drug which has arrayed proportional to the amount of drug in the blood stream at any time. P So we are also given that a physical 20 Q is equal to a few knots and the three piece equals three que easy quota point when you&#x27;re not talking. So we are required to get you off B intersil for the among off drug in the blood stream at peace equal to six if you&#x27;re not a sequel to 100 milligram. So for our solution, we can represent you over. The V is equal. Do now I got you King fue where in K is the proportionality constant So integrating this will have Vic you over Q physical toe the negative gave BP and getting the integral of both sides will have. And in a few music was the negative. Katie, let&#x27;s see. So today, seeing um, getting the power of the of both sides will have e off. Ellen of you is equal toe. Erase the negative. Katie, Let&#x27;s see. So remember here that sees disa call center, it can take up any value and our final and are resulting equation is Q is equal to see here days to the negative, baby. So this is our general solution. So we want to find the body off, See, nt and we do this making use of the initial conditions even in the problem. So AT T is equal to zero. It was said that you is equal to appear not so you&#x27;re not is a well toe See theories in the negative k, thank zero. So this will just cancel out the one therefore C is equal. Look, you&#x27;re not a resulting equation. Then Sq is equal to you&#x27;re not theories to the negative, Katie. So now we want to find the body off key. So a physical good three. We know that Q is equal to the point. Wendy off. You&#x27;re not to substituting this over equation will have point when you&#x27;re not, is equal to fear not. Ian is the negative gay 93 to be suggest Cancel out and we are left with 0.20 sequel To erase to the negative three K and solving Porky, we would have zero point 54. So reviving the Holik we should que off b is equal to. You&#x27;re not e eight days in the negative 0.54 feet. So we&#x27;ll use these to solve the problem in that beat for leather be. We have the sick with the seats and you&#x27;re not physical to 100 meaning it up. So computing for Q Off six, we have 100 erased in the negative 0.54 times six, which is approximately equal to 3.92 Milica. So this is our final answer.

Cyrielle Lorio · Answer

Okay, So suppose we have a drug here which enters the body at the read at 43.0 kilograms and lives above the other three proportional. The the amount of drugs in the body with a proportionality constant is equal to 0.2. So we&#x27;re also give in and Silicon Beach on at the is equal to zero. U is equal to zero. So for our So you, Sean, we can express this problem as the U over the P is equal to the rate in minus that it out. This is also equal to the U over dp as equals the 43 point boom miners 0.0 to you. So we want to express this problem in such a way that the proportional because that it&#x27;s outside the parenthesis of the agree shown. So we do this by isolating it, and we now have the view over the be as it was the negative 0.8 times u minus 526.83 So this value here is also equal to our equilibrium solution are Q bar. So this is now a separable equation so we can rearrange this equation X and expresses us pick you over Q Finals 500 went 6.83 in sick. Well connected the 0.8 the feet. So integrating both sides we now have and and off u minus 526.83 is it was the negative. Insert a 0.0 in pink. Let&#x27;s see. So we want to get the power Opie of both sides and we do this by no being a race. So l end of u minus 506.83 is equal to the series. The negative 0.8 booty. Let&#x27;s see. So we know that the left side will just be equal. Took u minus 500 going 6.83. And the sea here is just constant, so it can pick up any value. Therefore, we can ride the right side of the equation as ce days the negative 0.8 to be so further isolating you on the left side, our final general solution would be few is equal to 5 26 point 83 plus ce place the negative 0.0. Oh, sorry. C e race to the negative 0.8 to be. So this is our general solution. So now we want to find our particular solution. And we do this by using the initial condition given, which is a key physical zero we have. Q is equal to zero. So substituting this over general solution we have zero is equal to 5 26 point 83 plus ce place to the negative 0.8 to 0. So this will just be equal to one. And we now have C Is it called the negative pipe 26.83. So we can write our general solution as you is a quality 5 26 14 83 times one minus series of the negative, 0.8 to be So this is our particular solution. So if we&#x27;re going to grab our acquisition will have this kind of grab. So suppose this is our cue, which is the equilibrium solution. So we&#x27;ll have something like this. This is speak. So the a concentration will increase until it reached the equilibrium solution or the few bar and remains fund Stan afterwards until it reached the equity rooms shot. So that is our final answer

Fuzail Shakir · Answer

all rights of the rate at which the drug is administered to the patient in 30 the survival rate after triage s f T is equal to zero point is equal to eat the power natives, You&#x27;re going to the current spot of drug in the blood stream. P zero is equal to 80 milligrams. The amount of the drug that remains after 24 hours is to use equal to 24. So plugging this into the formula we can rewrite. This is 80 multiplied by E to the power off negative 0.2 multiplied by 24 plus the integral from 0 to 24 of s off 24 minus t quantified by our up t T t. So playing this in and simplifying the end up that this is equal to 0.6584 plus the integral from 24 0.6584 0 to 24 of E to the power of negative 0.2 multiplied by 24 linus t. This is multiplied by 30 uh, the teeth. So this is 0.6584 plus 30 multiplied by five e to the power of 0.2 t 0.2 tea minus 4.8. And we&#x27;re gonna evaluate this from 4.8 0 to 24. And if you plug in numbers, this is equal to 149.4 milligrams.

Stephen Hobbs · Answer

in this problem. We want to find the initial value solution, given the amount of drug and somebody&#x27;s system over time. And the first thing that we could do is rewrite this too. Why? Since we know that this differential equation is and linear form its original function must be an exponential of the form. Um, why TV amount of drug and the person system and milligrams over time, um is going to eat. We&#x27;ll see e to the K, which is negative 0.2 to on minus be just just three over K. Notice that we have a minus and another minus here. So the right side is going to simplify to positive 150. We&#x27;re then going to use the initial value over here to replace the left side with zero equal to, And we&#x27;re going to solve bird that see value, see, But at time, zero it&#x27;s gonna be e to zero and then plus 150. Either the zero is gonna cancel and so see is actually just gonna be minus or negative 100. And so finally, we&#x27;re gonna go ahead and replace, See with that so we could rewrite the initial value. We&#x27;re part one. A number graphic very shortly after is negative. 1 50 We need to know that value. We know he to OK you to the negative 0.2 times T and by annoyed. Plus one thing, if you want to double check this solution, you could see that e to the zero would be one. So we have negative 1 50 plus 1 50 which would cancel and get us back to zero. Which is true, because in the beginning of this problem, the patient does not have any of this drug in their blood stream yet. And then to graph gets in space to do so. Yeah. Um, we&#x27;ve got why of tea on the Y axis and exit. And, uh, the X axis is just gonna be time and really toe graft this Well, we should actually just go ahead and solve part be here, so we know that we&#x27;re gonna start, uh, zero. But part B is asking us. Then we&#x27;ll come back and graph it with the steady state solution is which is setting the differential equation equals zero, which, if we go back, that is the given information, right? Negative. Zero 0.2 Why? Plus three. And if we subtract three over and then divide, it will get negative. Three divided by a negative 0.2 but then give us a wide value of 150. And this is the steady state solution, um, which has a number of meanings. But I think one of the best is that it acts as a Nascimento. In other words, So 100 value 150 milligrams if you go ahead and draw, uh, direction field with a slope of zero. And, um, the amount of drug is going to then level off at that amount. Um, I think earlier I was saying minutes, but time is in hours on this particular problem here. So just a note on that and then lastly for part C, we want to find out. When do we reach 90% of this study state? Well, we just found out what the steady state was, which is kind of it&#x27;s Max among value there. And so if we multiply that by 0.90 that will tell us what 90% of it is, which ends up being 1 35 by the way, just gonna ride it out so you can see it. Um, we want to know how long this takes. Right? So you could actually just go ahead and, um, set this equal to the rest of the equation. Negative 1 50 You the negative hero boy to plus 1 50 Now, you could do this algebraic, lee, um, by solving for ah t here, which we&#x27;ve done in this problem in similar problems, a different way to think of it is, if you are at 135 let&#x27;s say grass like y equals 1 35 Then you want to find out when that hits the original equation, and that ends up happening at a time of 100 and 15. Again, I apologize. I can&#x27;t remember if it&#x27;s a minute, sir, hours here. But that will be our final amount of time before we reach 90

Kurt Kleinberg · Answer

Hi. We&#x27;re here to solve this differential equation with an initial condition in the context of intravenous drug injection were currently in the bloodstream. There is zero amount of the drug. So we&#x27;re gonna solve this. We&#x27;re gonna look at the graph. We&#x27;re gonna enter a few more questions about it as well. So recall that this differential equation is in the form of why prime A T equals K y plus B that the general solution to this is gonna be Wyatt T equals some constant times e to the k T minus B over K, so we can go ahead and plug in a bunch of stuff. Now we can plug in the initial value and Arby&#x27;s in our case. And so because our initial time zero this it just could be raised to zero minus R B is three over a negative 0.2 So we&#x27;re gonna be left with then at zero equals C plus 1 50 My apologies. This is gonna be as a plus 1 50 So negative 1 50 equal c. So our equation becomes why of tea equals negative 1 50 He raised to the negative 0.2 t plus 1 50 That graph looks like that. And you see that green line there 1 50? Because the next question is, when does the amount of the drug level off in your bloodstream Or in other words, what is the steady state of the drug? We can look at this two ways. One is as as t goes to infinity. This is important as two years to infinity. This is actually going to approach zero because it&#x27;s gonna be raised to some really, really big negative number which is one over E raised some huge positive number, which is basically zero. So this whole values essentially going approach zero, which means the level of the medication will approach 1 50 In fact, this will always be your ass until there, so it&#x27;s t goes to infinity. Why a t is gonna equal 1 50 Another way of looking that is, this needs have a net value of zero. The rape has to have a net value of zero four to level off so that you&#x27;re tangent is horizontal. And if you ask yourself negative 0.2 of something, when does that equal negative or ones that equal three part me negative three, because we&#x27;re gonna something to go each other. If you solve this, you get Y equals 1 50 And that&#x27;s why I have this line graft this green line right here to show that the graph does level off at 1 50 The next question is, when are we gonna have 90% of the steady state? So in other words, in other words, when is when are we 90% of the city state? Will that be 90% of 1 50? So that would be 0.9 of 1 50 Or, in other words, 135. So we have to solve this equation. 1 35 equals negative 1 50 b to the negative point. Oh, to t plus 1 50 Well, I&#x27;ll go ahead and subtract 1 50 Get this negative. 15 equals negative. 1 50 e to the negative 500.2 t will divide by negative 1 50 which is going to get us 1/10. We&#x27;ll go ahead and natural log bull science you get the natural log of 1/10 equals negative point. Go to tea. Well, the divide by negative point to and that is the time in which you have 90% of your study seat of the drug. This value when you calculate it, see your natural log one divided by a negative point to gets you roughly 115. Okay, so it&#x27;s gonna take you about 115 hours for this drug to reach 90% of its steady state.

Problem

For the following problems, set up and solve the …

Problem 286 Easy Difficulty

Answer

$y=\frac{r}{d}\left(1-e^{-d t}\right)$

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Gilbert Strang,

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Integration Review - Intro

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$y=\frac{r}{d}\left(1-e^{-d t}\right)$

Calculus

Catherine R.

Kayleah T.

Samuel H.

Joseph L.

Integration Review - Intro

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Gilbert Strang,Calculus

Catherine R.

Kayleah T.

Samuel H.

Joseph L.

Gilbert Strang,

Calculus