I think the answer is probably no, but not for the reasons other answers give. First of all we can ignore the whole multi-body problem: it's a really good approximation that the planets & Sun run on rails since they are hugely more massive than the spacecraft. let's also assume that modelling a trajectory between two points is tractable, whether or not you use continuous thrust or not (this could well be a reasonably hard optimisation problem to minimise fuel &c but I suspect that's very doable on a modern personal computer.
That's not what makes it hard: what makes it hard is that this is a search problem merely dressed up as a physics problem, and search problems, famously, have combinatorial explosions. Search problems require machines like Deep Blue to solve them, and these things are definitely supercomputers (albeit specialised ones).
Why is is a search problem? Well, because the way you get around the Solar System isn't in fact by computing a trajectory between two points, it's by computing a bunch of gravitational slingshots around other bodies in the Solar System. And there are a large number of such possible trajectories, and the number increases, possibly exponentially, as you increase the number of slingshots. And you can't deform the trajectories into each other to use any nice numerical solving approach because you keep crashing into planets since all these trajectories go rather close to planets.
Checking a proposed trajectory is much easier: if I tell you the plan is to do a couple of assists around Venus, a course correction burn in deep space then an assist around Earth and one around Jupiter on the way to Saturn (this is what Cassini did) then you can pretty easily check the trajectory is OK and compute its fine details. But arriving at such a trajectory is a different question. This smells strongly of P and NP: given a solution it is easy to check, but arriving at a solution might be hard.
So this might actually be a computationally seriously demanding problem. I think it probably isn't in fact, for a few reasons: there aren't very many objects you can use for slingshots so the search space doesn't explode too badly, and the mission duration is constrained as is fuel for course adjustments &c so you can prune solutions which take more time than you have or may need more fuel than you have. I suspect that keeps the computation sane.
[Note I'm posting this answer as a guest: I started writing it on the physics SE last night but the question got migrated & I don't belong to this SE.]