Assuming the definition of 'Closed Form' given in the table of: Closed Form Wikipedia entry, what areas tend to have problems that are traditionally expressed in closed form?
EDIT: Given the comment below about only 'easy' problems having closed form solutions, does this change if analytic closed forms are considered?
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The definition of a closed form solution, in particular what people agree to call an 'elementary' function, is culturally determined. A certain solution to a problem might be expressed as a series, or an integral, or some other limit, until its frequent use makes it worthwile having a standard name. That is how natural logarithms became accepted as 'elementary'. And depending on your field of specialization, you might or might not accept spherical harmonics as elementary functions. My personal favourite is the "hyperradical" function that maps a real number $a$ to the unique real root of the polynomial $x^5+x+a.$ The general fifth-degree polynomial is solvable in terms of algebraic operations plus the hyperradical function.
The original encyclopedia of Diderot and d'Alembert describes the debate about the definition of the word 'curve'. Traditionally, so goes the article, only algebraic equations are said to describe curves; but that definition does not satisfy people who are interested in, say, spirals. We (i.e., presumably d'Alembert) tend to subscribe to that point of view and accept differential equations to define proper curves, as well. Substitute the word 'elementary function' for 'proper curve' and you have a very similar debate.
My answer to your question, then, is: problems and solutions are traditionally expressed in closed form if and only if a sufficient number of people are interested in them for a sufficiently long period. You might summarize that in the adjective 'easy' but to me it does not cover the entire story.
By the way, the Wikipedia article is heavily biased towards calculus, but similar problems of definition arise in discrete mathematics.
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Closed-form solutions are in contrast to pure numerical solutions.
Closed-form solutions are of importance in applied mathematics, non-mathematical sciences and technology. Frequently recurring functions got a symbol and a name - the special functions. Often, a special function is solution of an equation of particular interest. Some arbitrary examples are
Closed-form solutions can help to discover relationships.
Closed-form solutions are the exception rather than the rule. They are droplets in an ocean of intractable computations.
In the frame of algebra/calculus, we have
Linear equations and systems of equations: yes, always (fortunately).
Algebraic equations and systems of equations: no (with a few exceptions, like second, third and fourth degree).
Transcendental equations: no.
Derivatives: yes, always.
Antiderivatives and summations: no (with exceptions such as rational fractions).
Definite integrals: no (a few dozen cases solved without antiderivatives are known).
Integral transforms (such as Laplace or Fourier): no (tables count a few dozen entries).
Linear differential equations and linear recurrences with constant coefficients: yes (but for the roots of the characteristic equation).
Linear or nonlinear differential equations: no.
Partial differential equations: no.
(When I say no, that means in general; closed-form cases do exist, but AFAIK they are sporadic and there is no easy way to characterize them.)
If you allow infinite sequences/series, presumably all these problems can be solved, by invoking numerical methods. At least in theory.
My definition of closed-form allows change of sign, addition, logarithm, exponential and function composition in the complex numbers. (Indirectly, subtraction, multiplication, division, exponentiation, polynomials, rational fractions, trigonometric and hyperbolic functions and their inverses.) This coincides with the Wikipedia entry, with the exception of the factorial.