What is the intuitive reason that partial differential equations are hard to solve? My question is why it is difficult to find analytical solutions for these equations.
Is there any intuition why numerical solutions are preferred?
 A: For differential equations in general I think there doesn't seem to exist a "method " for finding the solutions, unlike taking a derivative,  for instance,  which involves evaluating a certain limit, namely  $lim_{h\to 0}\frac {f (x+h)-f (x)}h $...  
A: Do not think that ODEs and PDEs are of different natures. An ODE is a kind of PDE wrt only one variable, while a PDE is wrt several variables.
It is not surprising that solving equations with more variables is more difficult than equations with less variables. I think that is the simplest intuitive reason which answers to your question.
Especially if one considers the boundary conditions in cases of ODE ("one variable PDE") and PDEs with more and more variables. 
Finding the general solution is one thing. But finding, among all solutions, the particular one which fits the boundary conditions is generaly an even more difficult task.
In case of "one variable PDE", the boundary conditions are given on a limited number of points (number related to the order of the equation). 
In case of "two variables PDE", the boundary conditions are given on a limited number of curves. In case of "three variables PDE", the boundary conditions are given on a limited number of surfaces. And so on.
One understand that the fitting task is harder in case of curve than in case of point, an even more arduous on surface. And so on.
