Is there some sort of list or table (like there is a table of integrals) that I could check my differential equation against whenever I encounter one?
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Even for the very basic case of solving for the primitive $y$ in $$dy=f(x)dx,$$ it is not very easy to determine in general when $y$ can be expressed as a combination of elementary functions alone. – Allawonder Oct 19 '20 at 16:22
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1The DLMF has many special functions and their related differential equations. The Wolfram functions site has similar lists. You can try Gradshteyn and Rhyzik also. – Somos Oct 19 '20 at 18:31
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What do you mean by a "special function"? There exist some kinds of differential equations that have solutions that can be written in terms of the "elementary" functions, polynomials, exponential, trig functions. There are far more kinds of differential equations that cannot be solved in terms those functions. We define a new function as a solution to such a differential equation and call it a "special function"- examples are "Bessel Functions", "Legendre functions", "Laguerre Polynomials" and others. There are, in fact, text books and lists of those but there can't be a list of "all" special functions because new "special functions" are being defined all the time.
user247327
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I meant the latter. Could you give me the name of one of those text book or lists? – Brain Stroke Patient Oct 19 '20 at 16:10
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I think what he means is, When can we say that a solution cannot be expressed in terms of elementary functions alone? – Allawonder Oct 19 '20 at 16:27
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I think what he means is, When can we say that a solution cannot be expressed in terms of elementary functions alone? – Allawonder Oct 19 '20 at 16:27