You should also think about what it means to be an exact solution. Many special functions that we use to write down "exact" solutions are defined through the differential equation that they solve. When you actually want to get a numerical, quantitative prediction, you need to evaluate the "exact" expression, which usually involves some iterative scheme, best performed through a computer. When you try to qualitatively understand the behavior of the solution, you use your knowledge of the exact function, say, through its asymptotic behavior, its extremal points, etc. 

Now we have a book-ful of special functions, which cannot possibly describe the solution space of all differential equations we can write. Especially nonlinear partial differential equations have a very rich solution space involving critical phenomena, solitons, etc. So how do you get information on equations that you cannot write a solution to?

One way to deal with this is to use abstract analysis. You analyze the equation on the basis of existence, asymptotic behavior, special solutions, etc. This allows you to understand its behavior to a certain degree.

Another way is to solve it numerically. Given initial and boundary conditions (depending on the type of your equation), you can see how the solution behaves by numerically solving the equation. This is not always possible, some equations are ill-posed, some break down due to formation of singularities. So don't think that numerics is an "easy way out". It's a complementary way to understand the space of solutions to equations when paper and pen are exhausted.

i think because there is no magic wand that would simply work on everything!, in short no exactness may exist!

Because firstly you can not find the exact solution to the complicated systems, you need to consider many simplifications, and secondly, its easier to program a numerical solution and integrate it in a model.

There is no general exact solution method for nonlinear ODEs and PDEs, so numerical approximate solutions is a must for most nonlinear systems. Though analytic solutions for linear ODEs and PDEs are established, solutions of some equations do not have a neat and easy-to-interpret closed form, e.g. infinite series, so numerical solutions are preferred for these systems, particularly for computer implementation 

You should also think about what it means to be an exact solution. Many special functions that we use to write down "exact" solutions are defined through the differential equation that they solve. When you actually want to get a numerical, quantitative prediction, you need to evaluate the "exact" expression, which usually involves some iterative scheme, best performed through a computer. When you try to qualitatively understand the behavior of the solution, you use your knowledge of the exact function, say, through its asymptotic behavior, its extremal points, etc. 

Now we have a book-ful of special functions, which cannot possibly describe the solution space of all differential equations we can write. Especially nonlinear partial differential equations have a very rich solution space involving critical phenomena, solitons, etc. So how do you get information on equations that you cannot write a solution to?

One way to deal with this is to use abstract analysis. You analyze the equation on the basis of existence, asymptotic behavior, special solutions, etc. This allows you to understand its behavior to a certain degree.

Another way is to solve it numerically. Given initial and boundary conditions (depending on the type of your equation), you can see how the solution behaves by numerically solving the equation. This is not always possible, some equations are ill-posed, some break down due to formation of singularities. So don't think that numerics is an "easy way out". It's a complementary way to understand the space of solutions to equations when paper and pen are exhausted.

In General for PDE, it very hard and impossible in some cases to obtain an explicit formula for the exact solution of a PDE , in addition if you have an exact solution for an equation and if you change the geometry of the domain , the solution change automatically. In practice the domaine are not smooth or let us say "classical" (like circles, rectangles...) but very complicated , and there is no explicit solution, for that we have to approximate the exacte solution by some numerical approximation. 

Most equations that model real-world situations are non-linear and do not have analytical solutions.  Therefore, one approach is to solve the equations numerically in order to obtain insights of the underlying system that is being studied.

"Exact" usually means a closed form solution in terms of elementary and special (usually) hypergeometric functions.  For this the symmetry must allow the pde to be separable.  In the nonlinear case, there are more problems.  

The simple answer is to remember how easy it is to differentiate versus integrate in closed form in 1D.  When you solve pdes you are integrating.  You can try perturbative methods and call that exact but you have to have some solutions in the linearized case to start with.  Even then, nonlinearities can get so large the perturbative methods fail as well.