Modelling Help needed

[rochus]

A drug is continuously introduced in a body over 4 hour period (same amount of drug enters body at any given time t).
Kidenys eliminate 2.5% of drug per hour. At end of 4 hours, the body needs 90 mg of drug.
A differential equation needs to be set up but I am not sure.
I thought

dD/dt = kt - m(0.025^t) where D=drug in body, k=drug entering body at any time t(constant?), t=time

we also know that when D=0, t=0 and that when D=90, t=4.
I need to find k.
Help please.

 

[galactus]

If we let r=rate it is entering.

Let k=rate it is leaving or being broken down.

A=amount in system at time t.

Then, we have the DE \(\displaystyle \frac{dA}{dt}=r-\frac{A}{40}\)

Use the integrating factor and use the initial conditions to solve.

Rewrite as \(\displaystyle \frac{dA}{dt}+\frac{A}{40}=r\)

Integrating factor is \(\displaystyle e^{\frac{t}{40}}\)

 

[rochus]

Thank you for the response.
However, could you eplain me why A is divided by 40 and where is the -2.5% an hour elimination rate in the differential equation?
Thank you.

 

[galactus]

2.5% is 1/40. I just used fractional form instead of decimals.

 

[a97775]

Hello, I'm interested in this thread as I have had difficulties in answering similar questions in the past.
I just wanted to know,
k=-A/40 ?
and isn't it constant?
If this were the case, after you use the integrating factor, would you get RHS = r*int(e^(t/40))dt ?