Term exact solution refers to some solution of certain equation(s), but also indicates some efficient way of the evaluation. Practically, the term "exact solution" has sense, indicating to some specific way of evaluation of solution, with is at least an order of magnitude more efficient than all other methods reported previously.
In particular, term "exact solution" may indicate conversion of the problem of some set to a problem from another set that is considered to be easier.
If deal with some ordinary differential equation, and the solution is expressed in term of quadrature (integral, where the integrand is expressed only in terms of already known, id est, spedial functions), then this expression can be qualified as exact solution.
If deal with a differential equation with partial derivatives, and some transform allows to express the solution in terms of an ordinary differential equation, such an expression can be qualified as "exact solution".
Also, the term "exact solution" may be used in the context of analysis of various approximations; while working with many various approximations, the specification "approximation" may be assumed as a default and omitted in the description. Then, the term "exact solution" indicates the solution a the background of approximations, even if no explicit way of the evaluation is provided.
In the most of cases, term "exact solution" refers to the representation of the solution, indicating some simple way of the analysis and evaluation, rather than to other properties of the solution.
http://prl.aps.org/abstract/PRL/v27/i18/p1192_1 Ryogo Hirota. Exact Solution of the Korteweg—de Vries Equation for Multiple Collisions of Solitons. Phys. Rev. Lett. 27, 1192–1194 (1971)
http://www.math.colostate.edu/~pauld/M546/Exact.pdf Paul DuChateau. INVERSE PROBLEMS IN PARTIAL DIFFERENTIAL EQUATIONS. 2. Exact Solutions. (2012)
http://web2.clarkson.edu/projects/crcd/me637/downloads/03_ExactSolutionsNSEq.pdf Goodarz Ahmadi. Exact Solutions to the Navier-Stokes Equation. (2012)
http://www2.mech.kth.se/~luca/Smak/rec5.pdf Fluid Mechanics, SG2214, HT2010 September 14, 2010. Exercise 5: Exact Solutions to the Navier-Stokes Equations I.