Homework Help: Differential Equations and growth constant

1. Feb 14, 2010

jofree87

dS/dt = kS - W

How do I solve this problem if k is the growth constant and W is also constant?

Here is what I have so far, but I dont think its quite right:

dS/dt = kS - W

(kS - W)-1dS = 1dt

1/k*ln(kS - W) = t + C

ln(kS - W) = kt + C1

kS - W = C2ekt

S = (C2ekt + W) / k

S = C3ekt + W/k

I think I did the math right but when I try plugging a few numbers in for the constant, the derivative doesnt match the function. Am I suppose to use the integrating factor for this problem?

2. Feb 14, 2010

ideasrule

But it does! Your solution satisfies the equation dS/dt = kS - W. Try plugging S in to that equation; you'll see that both sides are equal.

3. Feb 14, 2010

jofree87

Plug S into what equation?

I just think its wrong because if I take the derivative of S = C3ekt + W/k, then I would get S' = kekt, which isnt S' = kS - W. The W isnt suppose to disappear.

4. Feb 14, 2010

vela

Staff Emeritus
The term kS contributes a +W which cancels with the -W, leaving just the exponential.