Differential Equation Model Let's say we have a question like: As the salt KNO3 dissolves in methanol, the number x(t) of grams of the salt in a solution after t seconds satisfies the differential equation $ x'=0.8x-0.004x^2 $. How does one find the maximum amount M of the salt that will ever dissolve in the methanol?

If one looks at $x'(t)$, one observes the following facts:

1.) $x'(0 \, \text{grams}) = 0$;

2.) $x'(200 \, \text{grams}) = 0$;

3.) for $0 < x < 200 \, \text{grams}$, $x'(t) > 0$; the salt is going _into_ the alcohol;

4.) for $x > 200 \, \text{grams}$, $x'(t) < 0$; the salt is coming _out_ of solution.

Note that, by (2), if $0 \, \text { grams} < x < 200 \, \text {grams}$ initially, $x$ can never increase beyond $200 \, \text{grams}$ since $x'(t) = 0$ there. So assuming we start with $x$ as in (3), the salt will continue, on the net, to _enter_ the solution, at an ever decreasing _rate_ as $x$ gets close to $200 \, \text{grams}$, so that $x(t) \to 200 \, \text{grams}$ as $t \to \infty$. Eventually, the amount of salt in solution will be indistinguishable from $200 \, \text{grams}$, but never more. The model predicts a maximum observable amount $x = 200 \, \text{grams}$, provided $x$ starts off as in (3).

Hope this helps. Cheers, and

_**Fiat Lux!!!_**