Growth rate: If the amount S grows at rate ds/dt, then

[Guest]

If the amount of a substance S(t) grows at a rate, ds/dt , that is proportional to the square of its difference from some maximal amount M.

1) Write a differential equation which models the above statement. Let the constant of proportionality = k.

2) Find the general solution to the differential equation.

This problem is totally confusing. The first part I understand: "S(t) grows at a rate, ds/dt." The part I don't understand: "is proportional to the square of it's difference from some maximal amount M.

I'm guessing it is some variation of:

dP/dt = kP(1 - P/K)

Maybe something like:

dS/dt = kS(1 - S/M)

It is probably really simple, but the wording of the problem is confusing me.

 

[skeeter]

If the amount of a substance S(t) grows at a rate, ds/dt , that is proportional to the square of its difference from some maximal amount M.

dS/dt = k(M - S)²

dS/(M - S)² = k dt

1/(M - S) = kt + C

M - S = 1/(kt + C)

S = M - 1/(kt + C)

at t = 0, S = S_o ...

S_o = M - 1/C

S_o - M = -1/C

C = 1/(M - S_o)

S = M - 1/[kt + 1/(M - S_o)]

S = M - (M - S_o)/[kt(M - S_o) + 1]

 

[Guest]

Ok, so to find the specific solution with S(0) = 1 , k=3 , and M=2 :

S= 2- (2-S0)/[3(0)(2-S0)+1]

Is S0 the beginning value 1?

S= 2- (2-1)/[3(0)(2-1)+1]
S= 2- (1)/(1)
S= 1

Is that correct, or is S=1 and i find S0?

 

[Guest]

Ok, sorry, i'm still a little confused here, the S0 is throwing me off.

What would the equation to find the specific solution with S(0) = 1 , k=3 , and M=2 look like?

S = M - (M - S0)/[kt(M - S0) + 1]

Am i solving to find S0?

 

[tkhunny]

I do not know why the function notation was dropped. Let's put it back.

S(t) is your function in terms of t (and any other parameters, 'k', "M", and \(\displaystyle S_{0}\), in this case)

\(\displaystyle S_{0} = S(0)\)

You should be solving for a specific FUNCTION. You already know the parameters.

 

[skeeter]

What would the equation to find the specific solution with S(0) = 1 , k=3 , and M=2 look like?

S = M - (M - S0)/[kt(M - S0) + 1]

S = 2 - (2 - 1)/[3t(2 - 1) + 1]

S = 2 - 1/(3t + 1)