PDEs are very hard to solve.
I heard it claimed that it's very lucky, if you're someone who does mathematical finance, that the Black-Scholes partial differential equation, when used to price a call option, has a closed-form solution. (Well, if you include the error function in "closed form".)
The payoff of the thing being priced gives the initial condition for the Black-Scholes PDE and for a simple perturbation of that, there is probably not such a simple solution. And options markets probably would have developed very differently if there hadn't been an exact solution, because solving such an equation numerically, while possible, is a lot slower.
A friend of mine who knows more about PDEs than I do said that, basically, the set of exactly solvable PDEs has measure zero. Of course this isn't a theorem, but the point is that people who study PDEs don't expect there to be exact solutions.