Numerical methods on the other hand produce approximate solutions in the form of discrete values or numbers. Algebraic quadratic equations can also be solved exactly, if solutions exist, leading in general to two solutions. Finding exact solutions to higher-order algebraic equations will not, in general, be a feasible task and numerical methods must be employed to find approximate solutions instead. For the case of two or more coupled non-linear algebraic equations, numerical methods are routinely used today to find approximate solutions. Ordinary and partial differential equations are in general much more difficult to solve exactly.
Exact solutions are possible only in very special circumstances.
Realistic mathematical models will consists of non-linear differential equations or sets of coupled equations, possibly to be satisfied on general domains and subject to complicated initial and boundary conditions. These equations may depend on time and on one, two or three space dimensions. In such circumstances, explicit exact solutions will remain the dream of the mathematician for the foreseeable future. The differential equations governing the behaviour of an inviscid gas, the Euler equations, have been known to scientists for centuries, but the exact solutions of these equations available today are only valid for very simple physical situations. Exact solutions, even for unrealistic cases, are also helpful in assessing numerical methods intended for use in more general situations.