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Liouvillian solutions of second order differential equation without Fuchsian singularities

  • Michihiko Matsuda (a1)

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Consider a homogeneous linear differential equation of the second order whose coefficients are rational functions of the independent variable x over the field C of complex numbers. We assume that the coefficient of the first order derivative vanishes:

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References

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[1] Hailperin, R. R. (formerly Roberts, R. M.), On the solvability of a second order linear homogeneous differential equation, Doctoral Dissertation, Univ. Pennsylvania, 1960.
[2] Kaplansky, I., An introduction to differential algebra, Hermann, Paris, 1957.
[3] Kolchin, E. R., Existence theorems connected with the Picard-Vessiot theory of homogeneous linear ordinary differential equations, Bull. Amer. Math. Soc, 54 (1948), 927932.
[4] Liouville, J., Mémoire sur l’intégration d’une classe d’équations différentielles du second ordre en quantités finies explicites, J. Math. Pures Appl., 4 (1839), 423456.
[5] Matsuda, M., Lectures on algebraic solutions of hypergeometric differential equations, Lectures Math., Dept. Math. Kyoto Univ., 15, Kinokuniya, Tokyo, 1985.
[6] Rehm, H. P., Galois groups and elementary solutions of some linear differential equations, J. Reine Angew. Math., 307 (1979), 17.
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Liouvillian solutions of second order differential equation without Fuchsian singularities

  • Michihiko Matsuda (a1)

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