"exact soluble model" or "exact solvable model", "analytic" or "analytical" solutions

In physical science and math, we encounter some models that can be analytically solved. This means that the properties of models are fully understood and determined by the analytical solutions.

In the literature, we will refer those models as:

exact soluble model

or

exact solvable model

or

exactly soluble model

or

exactly solvable model

Q1. I wonder whether the usage of exact, exactly, soluble and solvable are interchangeable?

Q2. When we say the solutions are analytical, are that both analytic solutions and analytical solutions are correct usage?

Solution 1:

• Q1: In the technical language of mathematics, 'soluble' is interchangable with 'solvable', the former is British usage, the latter American. In chemistry, 'soluble' means 'can be dissolved in'.

  'exact' is an adjective modifying 'model', 'exactly' is an adverb modifying 'solvable'. So mathematically these are two distinct things (with the assumption that 'exact' has a stipulated mathematical definition. In informal English, these are hardly distinguishable, especially since it is becoming more common nowadays to drop the '-ly' (or put differently, it is becoming more common for the adjective form to be used adverbially).

• Q2: 'analytical' is more common than 'analytic' in standard English (both meaning roughly the same thing 'able to think clearly and precisely'), but in math 'analytic' is a technical adjective form with a very precise meaning. The only instance I know of for 'analytical' is in the phrase 'analytic continuation'. There may well be other instances, but 'analytical continuation' is all I know of.