calculus 2 homework help?

Question

Suppose we know that the amount of a substance S grows at a rate of proportional to the square of its difference from its maximal amount M.

a) write a differential equation which models the situation.
b) find a general solution to the differential equation.
c) find the specific solution satisfying s(0)=1, if the constant of proportionality is 3 and M=2

mcbengt · Accepted Answer

Any equation of the form

dS/dt = k (S - M)^2

with k and M constants, models this situation.  (M is the maximal amount, and k is the constant of proportionality between dS/dt and (S - M)^2).

This is a separable differential equation.  Using the standard method for solving it, you can write it as dS/(S - M)^2 = k dt, and then integrate both sides to learn learn that -1/(S - M) = kt + C for some constant C, which implies that -1/(kt + C) = S - M, which implies that

S = M - 1/(kt + C)

for some constant C.

If we take k = 3 and M = 2 then we have

S(t) = 2 - 1/(3t + C)

and since S(0) = 1 we must have 1 = 2 - 1/(3*0 + C), which implies that 1 = 2 - 1/C, which implies that 1/C = 1, or that C = 1, so that

S(t) = 2 - 1/(3t + 1)

for all t.

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calculus 2 homework help?

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