On the Euler-Imshenetskii-Darboux transformation of linear second-order equations
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It is shown how the linear Euler-Imshenetskii-Darboux (EID) differential transformation can be used for generating infinite sequences of linear second-order ordinary differential equations starting from certain standard equations. In so doing, the method of factorization of differential operators and operator identities obtained by means of this method are used. Generalizations of some well-known integrable cases of the Schrödinger equation are found. An example of an integrable equation with the Liouville coefficients, which apparently cannot be solved by the well-known Kovacic and Singer algorithms and their modifications, is constructed. An algorithm for solving the constructed class of equations has been created and implemented in the computer algebra system REDUCE. The corresponding procedure GENERATE is a supplement to the ODESOLVE procedure available in REDUCE. Solutions of some equations by means of the GENERATE procedure in REDUCE 3.8, as well as those obtained by means of DSOLVE in Maple 10, are presented. Although the algorithm based on the Euler-Imshenetskii-Darboux transformation is not an alternative to the existing algorithms for solving linear second-order ordinary differential equations, it is rather efficient within the limits of its applicability.
KeywordsDifferential Equation Operating System Artificial Intelligence Integrable Equation Ordinary Differential Equation
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- 1.Euleri, L., Methodus Nova Investig andi Omnes Quibus Hans Aequationem Differentio-differentialen ∂∂y(1 − axx) − bx∂x∂y − cy∂x 2 = 0, M.S. Academiae Exhibuit die 13 Ianuarii 1780, Institutiones Calculi Integralis 4, 1794, pp. 533–543.Google Scholar
- 2.Imshenetskii, V.G., Extension of the Euler Method for Studying Integrability of One Particular Class of Linear Second-Order Equations to the General Case of Linear Equations, Zapiski Imperatorskoi Akademii Nauk, St. Petersburg, 1882, vol. 42, pp. 1–21.Google Scholar
- 6.Whiting, B.F., The Relation of Solutions of Different ODE’s Is a Commutation Relation, Differential Equations, Knowles, I.W. and Lewis, R.T., Eds., Horth-Holland: Elsevier, 1984, pp. 561–570.Google Scholar
- 7.Gradshtein, I.N. and Ryzhik, I.M., Tablitsy integralov, summ, ryadov i proizvedenii (Tables of Integrals, Sums, Series, and Products), Moscow: Nauka, 1971, 5th ed. [New York: Academic, 1980].Google Scholar
- 10.Hearn, A.C., REDUCE: User’s Manual, Version 3.8, http://www.reduce-algebra.com/documentation.htm.
- 11.Neun, W., REDUCE. User’s Guide for Personal Computer, Berlin: Konrad-Zuse-Zentrum, 2000.Google Scholar
- 12.www.maplesoft.com/products/maple/manuals.Google Scholar
- 13.Singer, M.F., Galois Theory of Linear Differential Equations, Grundlehren der Mathematischen Wissenschaften, vol. 328, Springer, 2003.Google Scholar