It's given that a model for seasonal growth is dp/dt = kPcos(rt - m) and asks to solve for the initial condition P(0)=P(subscript0). I wasn't sure how to make a subscript so to make it easier lets just say P(0) = P*. I'm not quite getting the answer that's in the book and I was wondering if someone could show me the steps so I can see what I'm doing wrong. The answer I got is almost what's in the book so I'm sure it's something small I'm overlooking/forgetting. This is what I got.
dp/kP = cos(rt - m)dt
Integrating both sides gives
ln|kP|/k = rsin(rt - m) + C
Multiplying out the k, putting both sides as a exponential of e and dividing by the k inside of the logarithm gives
P = e^krsin(rt - m)e^kC/k
if we let e^kC/k = A we get
P = Ae^krsin(rt - m)/k
Solving for the initial condition P(0) = P* gives
A = P*k/e^krsin(-m)
Plugging that into P gives
P = P*e^krsin(rt -m) - krsin(-m) or P = P*e^kr[sin(rt - m) + sin(m)]
The only difference is the book has k/r in the exponent on e not kr and I'm not sure why.
dp/kP = cos(rt - m)dt
Integrating both sides gives
ln|kP|/k = rsin(rt - m) + C
Multiplying out the k, putting both sides as a exponential of e and dividing by the k inside of the logarithm gives
P = e^krsin(rt - m)e^kC/k
if we let e^kC/k = A we get
P = Ae^krsin(rt - m)/k
Solving for the initial condition P(0) = P* gives
A = P*k/e^krsin(-m)
Plugging that into P gives
P = P*e^krsin(rt -m) - krsin(-m) or P = P*e^kr[sin(rt - m) + sin(m)]
The only difference is the book has k/r in the exponent on e not kr and I'm not sure why.
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