Never mind. I think I figured out what I did wrong. I wasn't supposed to dived by P. I needed to divide by P-a and do a u-substitution
The problem says, "Solve the differential equation. Assume a, b, and k are nonzero constants."
dP
dt = P - a
Here is how I tried to solve the problem:
dP
dt = P - a
multiply both sides by dt:
dp = P - a dt
divide both sides by P:
1/P dP = -a dt
take integral of both sides:
(integral sign) 1/P dP = (integral sign) -a dt
ln|P| = -at + C
exponentiate both sides to get rid of natural log:
e^(ln|P|) = e^(-at) * e^(C)
P = Ce^(-at) <----This is my answer, where C is just a constant.
However, my book says that the answer is P = a + Ae^(t). Please let me know what I did wrong. Thank you