I have an equation related to the Grompertz Growth Curve that has the form:
$$ \frac{dP}{dt} = k_L P(t)(M - P(t)) $$
$K_L = \frac{1}{70} $ and $M = 100$
re arranging this we have
$$ \frac{dP}{P(t)(M - P(t))} = \frac{1}{70} dt $$
L.H.S. can be re written as
$$\frac{dP}{P(t)(M - P(t))} = \frac{A}{P(t)} + \frac{B}{(M - P(t))} $$
From this we can see that $A = \frac{1}{100} \rightarrow AM = 1, A = \frac{1}{M}$
and $-A + B = 0 $ which implies that $ B = A $, so $ B = \frac{1}{100} $
We now have two separate equations, but I am running into a problem. How do I integrate a function?
I have
$$ \frac{1}{100} \bigg( \int \frac{dP}{P(t)} - \int \frac{dP}{ P(t) - M } \bigg) = \frac{1}{70} \int dt$$
This is where I get lost and do not know how to do it. I know that the integral of $\frac{1}{x}$ is ln|x|, but I do not see how that helps me as I have a function in the denominator rather than a variable. Any help would be greatly appreciated!
If the integrals were with respect to time, you were stuck, but the variable of integration is the function $P$ itself, so you can use the rule of $ \int \frac{1}{x} dx = \ln |x| $
$$ \ln P - \ln (P-M) = \frac{10}{7}t + c $$ $$ \ln \frac{P}{P-M} = \frac{10}{7}t + c $$ $$ \frac{P}{P-M} = e^c \cdot e^{\frac{10}7 t} = A e^{\frac{10}7 t}$$ $$ P = A e^{\frac{10}7 t} P - A e^{\frac{10}7 t} M $$ $$ P (1 - Ae^{\frac{10}7 t}) = -MA e^{\frac{10}7 t}$$ $$ P = \frac{-MA e^{\frac{10}7 t}}{1-A e^{\frac{10}7 t}} $$