ODE's: Rate of Change for Drug Dissipation in Human Body

[Leptos]

Homework Statement

Suppose that the human body dissipates a drug at a rate proportional to the amount y of drug present in the bloodstream at time t. At time t = 0 a first injection of Y₀ grams is made into a body that was drug free prior to t = 0.

a) Amount of drug at the end of T hours.
b) If at time T a second injection of y₀ is made, find the amount of drug at the end of 2T hours.
c) If at the end of each time period of length T an injection of Y₀ is made, find the amount of rug at the end of n*T hours.
d) Find the limiting value of the answer to (c) as n->∞.

Homework Equations

dy/dt ∝ y(t)
y(0) = y₀

The Attempt at a Solution

I'm not sure what equation would give me the model required for this question...
It appears we have logistic decay so it would be something like:
y₀e^-kt/(1 - e^-kt)
I'd appreciate being nudged in the right direction.

 

[Leptos]

Well can someone at least tell me if I'm on the right track? Would I use logistic decay to model this situation?

 

[Mark44]

Homework Statement

Suppose that the human body dissipates a drug at a rate proportional to the amount y of drug present in the bloodstream at time t. At time t = 0 a first injection of Y₀ grams is made into a body that was drug free prior to t = 0.

a) Amount of drug at the end of T hours.
b) If at time T a second injection of y₀ is made, find the amount of drug at the end of 2T hours.
c) If at the end of each time period of length T an injection of Y₀ is made, find the amount of rug at the end of n*T hours.
d) Find the limiting value of the answer to (c) as n->∞.

Homework Equations

dy/dt ∝ y(t)
y(0) = y₀

The Attempt at a Solution

I'm not sure what equation would give me the model required for this question...
It appears we have logistic decay so it would be something like:
y₀e^-kt/(1 - e^-kt)
I'd appreciate being nudged in the right direction.

This sentence,

Suppose that the human body dissipates a drug at a rate proportional to the amount y of drug present in the bloodstream at time t.

translates into dy/dt = ky, for 0 <= t <= T. Since the drug dissipates over time, k has to be negative.
Find the solution to this differential equation, using the initial condition.

At time T, another dose of the drug is injected, so come up with a differential equation that represents this situation for T <= t <= 2T.

 

[Leptos]

This sentence,

translates into dy/dt = ky, for 0 <= t <= T. Since the drug dissipates over time, k has to be negative.
Find the solution to this differential equation, using the initial condition.

By separation of variables I get y = Ce^kt where C = e^c and then y(0) = y₀ = Ce₀ so y = y₀e^kt.

At time T, another dose of the drug is injected, so come up with a differential equation that represents this situation for T <= t <= 2T.

Would this be shifting the coefficient on e^kt? So in this case we would have y(T) = y₀e^kT therefore we use y = (y₀e^kT)e^kt. Is this right?

 

[ksallegue]

can i see the solution for this problem? it is also one of my problem.