Unbelievably, a lot of people believe they can.
I don't think their models truly use differential equations. They have fudge factors, adjusting constants, and trend altering factors to "fit" the "historical" cherry picked and falsified data added to the differential equations to steer the results.
How any sane and technically knowledgeable person could believe the future predictions of such a program is beyond me. GIGO
The way to predict short term differential equations is using numerical methods. Runge-Kutta 4th order is one well known method. Adams-Moulton 4th order predictor-correctors are another that has a smaller error, at the cost of needing to run the predictor to get the next point, and the corrector to adjust the point to get the small error term. Long ago we worried about the error term because the number of function evaluations (cost) was inversely related to step size, and a small error let you get equivalent error with a larger (cheaper) step size.
Another approach which they do use is to linearize the nonlinearities. That way your predictions become repeatable, and predictable, but has the minor difficulty that the problem you are solving is not the one you were given. The moral equivalent of multiplying by zero and adding what you hope is the right answer.