Let f(t) be the weight (in grams) of a solid sitting in a beaker of water. Suppose that the solid dissolves in such a way that the rate of change

Question

Let  be the weight (in grams) of a solid sitting in a beaker of water. Suppose that the solid dissolves in such a way that the rate of change (in grams/minute) of the weight of the solid at any time  can be determined from the weight using the forumula: If there is 4 grams of solid at time  estimate the amount of solid 1 second later.

Answer 1

To solve this the differential equation f'=-4f(3+f) must be solved. It is separable into df/(-4f(3+f))=dt allowing direct integration of both sides. For the left side a partial fractions method can be used to modify it into +df/12f-df/(3+f) and the right side needs no modification. Completing the integrals yields 1/12(ln(f)-ln(3+f))=t+C which can be simplified and C evaluated using f=4 when t=2 (given) to be f=3/(e^12t-23.4 -1). Thus f(1)=-3 which is nonsense telling us that the model breaks down near f=0