It's a pretty open-minded exercise I found online. It says, you're advising a social network company and they're trying to model an equation for $u(t)$, being this the amount of active users in the network.
They know their equation will have this following "shape". $$\frac{\partial^2 u}{\partial t^2} + c_{1} \frac{\partial u}{\partial t} + c_{2} (1 + t)u = K$$
It says, how would you gather the data to make research easier (seems you have that social network, or a similar one you can sample data from). What mathematical tools would you use to estimate the coefficients $c1$, $c2$, and $K$. Finally, how would you verify and validate the final model you'd obtain?
If the experimental data $(u_1,t_1)$ , $(u_2,t_2,)$ , ... , $(u_k,t_k)$, ... , $(u_n,t_n)$ are sufficiently numerous and well distributed, it is possible at each $t_k$ to compute approximates of $\frac{du}{dt}$ and $\frac{d^2u}{dt^2}$ thanks to numerical methods. Then a simple linear regression will lead to approximates of $c_1$ , $c_2$ and $K$.
But it is not the more accurate method, especially in case of not regulary distributed $t_k$. Better use numerical integration and the equation : $$u +c_1\int udt +c_2\int\left(\int (1+t)udt \right)dt=K\left(\frac{t^2}{2}+at+b\right)$$ A linear regression leads to $c_1$, $c_2$, $K$, $a$, $b$.
The additional constants $a$ and $b$ depend on the lower bound of the integral in the numerical process. They are not used latter.
The general principle of this method is explained in the paper : https://fr.scribd.com/doc/14674814/Regressions-et-equations-integrales
Of course, this would require a few preliminary studies and tests in order to develop a reliable algorithm. If you need more specific information and help about this method, you are welcome to contact me directly.