Integration of differential equations in closed form

The representation of solutions of differential equations by algebraic formulas, using an a priori given supply of functions and pre-given mathematical operations.

If one takes as the functions: the elementary functions plus the functions participating in the equation; and as the operation: finite sequences of algebraic operations plus the operation of taking the indefinite integral (quadratures) of the functions given, then one speaks of the integration (i.e. solution) of differential equations by quadrature. The Bernoulli equation is an example of an ordinary differential equation that is integrable by quadrature. J. Liouville (cf. [1]) was the first to indicate an equation that could not be integrated by quadrature. Thus, the solution of

$$y''+xy=0\label{*}\tag{*}$$

cannot be expressed by elementary functions and integrals of them (cf. [2]). Ordinary differential equations can only extremely seldom be integrated by quadrature. Most profound results on the possibility of integration by quadrature have been obtained on the basis of S. Lie's theory of continuous transformation groups (cf. [3], [4], [5]). Special functions, representations of solutions by convergent series, etc., are allowed for when integrating differential equations in closed form. Any concrete linear differential equation with variable coefficients is integrable in closed form if one includes a finite number of (usually special) functions making up a fundamental system of solutions in the supply of admissible functions. E.g., if one introduces the Bessel functions, then the general formula for the solution of \eqref{*} can be given in terms of them. Thus, differential equations are a source of special functions, and the inclusion of special functions in the supply of admissible functions allows one to enlarge the class of equations that are integrable in closed form. However, the problem of integrating non-linear equations in closed form does, in general, not reduce to complementing the set of formulas by a finite number of special functions.

Formulas for the solutions of partial differential equations can be obtained only in individual very special cases (see, e.g., d'Alembert formula). Group methods (cf. [5]) have great significance in finding formulas for solutions of such equations.

References

 [1] J. Liouville, J. Math. Pures et Appl. Sér. 1. , 6 (1841) [2] I. Kaplansky, "An introduction to differential algebra" , Hermann (1957) [3] S. Lie, G. Scheffers, "Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen" , Teubner (1891) [4] E.L. Ince, "Ordinary differential equations" , Dover, reprint (1956) [5] L.V. [L.V. Ovsyannikov] Ovsiannikov, "Group analysis of differential equations" , Acad. Press (1982) (Translated from Russian) [6] E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , 1. Gewöhnliche Differentialgleichungen , Akad. Verlagsgesell. (1943) [7] E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , 2. Partielle Differentialgleichungen erster Ordnung für eine gesuchte Funktion , Akad. Verlagsgesell. (1944)