Working through a homework problem for my differential equations course. We're modeling the time it takes to cure a staph infection based on a dosage of antibiotic. We're given the initial value problem $x'=0.02310x - 0.01d$ where $d$ is dosage of antibiotics in mg. The initial condition is $x(0)=1$ Thus I've constructed the solution formula as follows: $$x=\left(1-\frac {0.01d}{0.02310}\right)e^{0.02310t} + \frac {0.01d}{0.02310}$$

This I believe is sound. The next question asks us to find a critical dosage $d$. We know by looking at approximations from slope fields that the critical dosage is somewhere between $1.5g$ and $3.0g$ however, I'm stuck as how to use my above equation to solve for the exact value for $d$ without having it in terms of some $t$. Can anyone push me in the correct direction? Or perhaps catch a previous mistake??

  • $\begingroup$ If you want to solve for $d$ and if your equation is correct, just use $x(0)=1$. $\endgroup$ – Git Gud Sep 12 '13 at 17:07
  • $\begingroup$ My mistake, I'll correct it quickly, our initial condition was x(0)=1, whoops. $\endgroup$ – Ben Anderson Sep 12 '13 at 17:10
  • $\begingroup$ I edited my comment. It's even easier now. $\endgroup$ – Git Gud Sep 12 '13 at 17:11
  • $\begingroup$ But then I just get a true statement, that is $1=1$ the $d$ goes away...? am I missing something obvious here? $\endgroup$ – Ben Anderson Sep 12 '13 at 17:16
  • $\begingroup$ No. What this tells you is that any $d$ will work. If that's odd, then maybe there's a mistake somewhere before that. Probably in problem's modelation. $\endgroup$ – Git Gud Sep 12 '13 at 17:18

More context, please...

What is $x$? Is it the percentage of infected cells as a function of time? That $x$ starts from $x(0)=1$ and its growth is slowed down by the administered antibiotic $d$ seems to point to that direction... But it can't be, since $x$ (eventually) becomes greater than one for "small" $d$ and negative for "large" $d$.

Also, what is a 'critical dosage'? There's no way to get $d$ from your solution + initial value, as it is part of the original model; that is just common sense...

Are you then looking for a value of $d$ which would send $x$ to zero? If you do, then the critical dosage is the value of $d$ making the coefficient of the exponential zero: for smaller $d-$values, your solution increases (starting form one) exponentially in time; for larger $d-$values, $x$ decreases, hits zero & then becomes negative.

Btw, it's not your fault of course, but that looks like a terrible, terrible model... Is it meant to only hold approximately for a finite period of time?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.