Are analytic solutions of differential equations realistic? I'm studying ordinary differential equations and learned some methods to solve linear ones. I'm definitely a novice. Well I think I can follow the procedures and math behind it. However, I'm not sure that differential equations actually model realistic phenomena. For example, a car reduces its speed continuously and finally stops in real world. Let's denote the velocity of the car as $v(t)$ which is a function of time $t$. And let $\tau$ be the time when the car stops. Then $v$ is not an analytic function. It is singular at $\tau$. We may assume that $v$ is infinitely differentiable at $\tau$ but anyway it is not analytic there. Can we model this movement of the car with a differential equation? All solutions of the differential equations I have learned were analytic. I think the equations themselves don't put such restrictions. But I don't get why the solutions are all analytic. Are there special methods to obtain non-analytic solutions? Or should the solutions of differential equations be analytic? If so, how can we model the movement of the car with differential equations? I have navigated my textbook but found no clue (The book may be too elementary).
 A: First of all don't confuse the possibility of modeling something with a differential equation with whether a particular differential equation models something. Basically what you have is a differential equation that doesn't capture some of the aspects of reality very well, this cannot be taken as argument that these aspects can't be modeled using a(nother) differential equation.
Note that most models are meant to be mere approximations, they will therefore in some cases deviate significantly from reality in some cases or at some aspects.
Also there's no real requirement that the functions in a differential equation is analytic, obviously there's no reason to assume that the solution is either. However it's often so in physics that one assumes that one can differentiate all functions, but strictly that might not be required anyway in the normal sense (one can use so called distribution to get around some situations where the function is apparently non-differentiable). 
As for the stopping car, there's no actual reason to believe that the movement of it isn't differentiable. Even if it looks like it comes to a sudden stop it could be that it's just that we don't look close enough, maybe the deceleration decreases quickly instead of just transiting directly to zero?
If we're to use the model that the braking force is constant whenever the car is in motion the solution for $v(t)$ would have a corner-point where it comes to a stop, but the model would as a differential equation be $v'(t)=F_0\operatorname{sgn}(v)/m$. We can handle this in various way, one is to rewrite it as an integral equation instead $v(t) = \int F_0\operatorname{sgn}(v) dt/m$ which won't be hindered by the corner-point. Another way is to use distributions where $v$ would be differentiable anyway. A third approach is to allow non-differentiability at isolated points where other requirements governs the solution (for example allowing $v(t)$ do be non-differentiable at isolated points given it's at least continuous there).
