Find the solution of the seasonal growth model.

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.

 

this might help to explain why is k/r:

https://en.zdam.xyz/problem/22891/

 

Oh lol. The r goes on the bottom cause after differentiation using the chain rule the r has to disappear to get back to the original expression meaning there has to be an r on the bottom. for some reason I just put it out front. I was mixing myself up thanks.