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
Favorite 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.