Problem: The rate of change of the total number of sales, $dS(t)\over dt$, of a new product is proportional to $ S(t)\over t^2$, where $t$ is in years. If the saturation of the market is $50000$ units, and if after $1$ year the total numbr of sales is $10000$ units, find the number of sales at any time $t$.
I have tried solving this problem by setting it up as follows
$$\frac{dS(t)}{dt} = k \frac{S(t)}{t^2}$$
I come to the answer of $S(t)$ = $ce^{-\frac{k}{t}} $ or $\ln(S(t)) = -\frac{k}{t} + \ln(c)$
This allows me to solve for $c$ getting $ \ln(c)= \ln(10000) + k$
However, even if I subsitute this back in for $c$ and try to solve for $k$ I am out of given data. Am I doing the problem wrong or what have I missed. I appreciate any help.
The saturation of the market is the total number of sales as $t\to\infty$. Taking this limit in your solution, we find $$ 50000=\lim_{t\to\infty}S(t)=c. \tag{1} $$ Now we can find $k$ by plugging $t=1$ in the equation $\ln(S(t))=-\frac{k}{t}+\ln(c)$: $$ \ln(10000)=-k+\ln(50000) \implies k=\ln(5). \tag{2} $$