Differential Equations and growth constant

[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 don't 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 = C₂e^kt

S = (C₂e^kt + W) / k

S = C₃e^kt + W/k

I think I did the math right but when I try plugging a few numbers in for the constant, the derivative doesn't match the function. Am I suppose to use the integrating factor for this problem?

 

[ideasrule]

I think I did the math right but when I try plugging a few numbers in for the constant, the derivative doesn't match the function

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

 

[jofree87]

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

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' = ke^kt, which isn't S' = kS - W. The W isn't suppose to disappear.

 

[vela]

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