Construct the matrix diagram for each of the following models. Drug diffusion:A drug taken orally eventually diffuses into the b
loodstream through the stomach.Let st be the amount of drug in the stomach at the times t and bt the amount that has been absorbed into the bloodsteam. each time step, 50% of the drug is absorbed from the stomach into the bloodstream and 80% of the drug in the blood stream is metabolized. we have st+1= 0.9st, bt+1=0.5st+0.2bt
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It looks like the differential equation is
Check for exactness:
As is, the DE is not exact, so let's try to find an integrating factor <em>µ(x, y)</em> such that
*is* exact. If this modified DE is exact, then
We have
Notice that if we let <em>µ(x, y)</em> = <em>µ(x)</em> be independent of <em>y</em>, then <em>∂µ/∂y</em> = 0 and we can solve for <em>µ</em> :
The modified DE,
is now exact:
So we look for a solution of the form <em>F(x, y)</em> = <em>C</em>. This solution is such that
Integrate both sides of the first condition with respect to <em>x</em> :
Differentiate both sides of this with respect to <em>y</em> :
Then the general solution to the DE is