It would have been easier to have a normal pdf and then use MLE , at least to get a good idea about the numerical solution.
In such situations, here is what you do. You assume the parameters, Beta0, etc. Then for each observation pair (x,y), plug in the formula and calculate L. Hence, you shall have 50 such L ( call them L1, L2, L3.... ). You can simultaneously take the log of these L's (Call them Log_L1, Log_L2, etc).
Then you add them up. Call the total Sum = Sum_LL.
Then you maximize the sum_LL by changing the parameters, beta0, beta1 and with any other constraints on the betas that you can think of to make the solution viable.
The key here is the first guess of the betas. Also, you might need an optimizer to maximize sum_LL. You can do this in excel (using solver) or in matlab. In summary it is an iterative procedure. 1) you assume parameters (betas) , 2) you plug (x,y) with the assumed betas in the Log function 3) Calculate sum of the Log Likelihood. 4) Change the betas in until there is no further increment in the sum of the Log LL (essentially optimize w.r.t betas).