# The difference between exact solution and analytic solution

My mother tongue is not English so I am confused with the difference between exact solution and analytic solution. Are these the same?

Yes, they are (for the most part) the same, typically being used interchangeably.

And really, both are kind of a misnomer, because things typically not considered "analytic" or "exact" forms are, in fact, very often "exact" in the literal sense. There is nothing "inexact" about, say, the expression

$$e = \lim_{n \rightarrow \infty} \left(1 + \frac{1}{n}\right)^n$$

for $$e$$, it's just that because it involves a limit, it is not consider as an "analytic" or "exact form" in the kind that is referred to by these terms. What these terms really mean is what would perhaps better be described as a "solution by special functions": we fix a finite number of "allowable" (or "special") functions and constants which (ideally) we understand well, and try to describe the solution of some given problem in terms of a finite composition of these. And when it is said that "no exact/analytic solution exists", it means that for some set set of such functions, we cannot write down a formula of this type for the given problem.

Of course, as you can tell, this depends very much on what special functions we allow. Most restrictively and commonly, these are just the so-called "elementary functions" consisting of addition, multiplication, exponentiation, and their inverses (trig. functions are also typically included but these are a special case of exponentiation when you use complex numbers). More "liberally", we may add some "non-elementary" special functions like the gamma function or Lambert's W-function to the mix, and depending on whether you do this, the range of problems for which you can give an "exact/analytic" solution will change.

That said, they are generally contrasted against numerical solutions which, by definition, are truly inexact or approximate. These solutions are basically amounting to generating a big list of approximate numbers through the use of a computer to specify the desired function over a range of points. They are used because they are extremely general: very little is not amenable to a numerical solution, though the computation may be extremely expensive. The downside besides not being exact (though we should remember that scientific models, in general, cannot be faithfully relied with certainty as being truly "exact" to the "real world" anyways), is that unlike a formula which can have parameters, they only ever describe one instance of a problem, they do not describe general instances. Hence, a new numerical solution needs to be calculated in each new instance we want to study, and as said, the computational expense can make this prohibitive.

• Thank you very much!! – nana Sep 27 '19 at 13:45