# Modeling with Linear Differential Equations

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??

• If you want to solve for $d$ and if your equation is correct, just use $x(0)=1$. – Git Gud Sep 12 '13 at 17:07
• My mistake, I'll correct it quickly, our initial condition was x(0)=1, whoops. – Ben Anderson Sep 12 '13 at 17:10
• I edited my comment. It's even easier now. – Git Gud Sep 12 '13 at 17:11
• But then I just get a true statement, that is $1=1$ the $d$ goes away...? am I missing something obvious here? – Ben Anderson Sep 12 '13 at 17:16
• 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. – Git Gud Sep 12 '13 at 17:18

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.