I have a calculus question that I can't figure out. Thank you in advance for the help!
A model for tumor growth is the Gompertz function that is a solution to the differential equation
$\dfrac{dy}{dt} = ay\ln\left(\dfrac{K}{y}\right)$ where $y$ is the weight of the tumor in milligrams, $t$ is measured in days, $a$ is a constant, and Upper $K$ is the limiting size of the tumor. Assume that $a = 0.8$ and Upper $K = 100$.
Find a solution to this differential equation that satisfies $y(0) = 1$ mg. enter the exact answer
$y(t)= ..$
Let $y = \frac{K}{u}$ so that
$$\frac{dy}{dt} = -\frac{K}{u^2}\frac{du}{dt}.$$
Your ODE becomes
$$\frac{du}{dt} = -au\ln u,\quad u(0) = K.$$
By separation of variables,
$$\int \frac{du}{u\ln u} = \int -a\, dt$$
or
$$\ln|\ln u| = -at + C$$
where $C$ is a constant. Hence
$$\ln u = Ae^{-at}$$
where $A$ is a constant. Invoking the initial condition $u(0) = K$, we obtain $A = \ln K$. Thus
$$u(t) = \exp\{e^{-at}\ln K\} = K^{\exp\{-at\}},$$
or
$$y(t) = \frac{K}{u(t)} = K^{1 - \exp\{-at\}}.$$