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Transactions of the American Mathematical Society

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Integration of ordinary linear differential equations by Laplace-Stieltjes transforms


Authors: James D’Archangelo and Philip Hartman
Journal: Trans. Amer. Math. Soc. 204 (1975), 245-266
MSC: Primary 34A25
DOI: https://doi.org/10.1090/S0002-9947-1975-0357935-8
MathSciNet review: 0357935
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Abstract: Let $ R$ be a constant $ N \times N$ matrix and $ g(t)$ an $ N \times N$ matrix of functions representable as absolutely convergent Laplace-Stieltjes transforms for $ t > 0$. The paper gives sufficient conditions for certain solutions of the system $ y' = (R + g(t))y$ to be expressed as Laplace-Stieltjes transforms or as linear combinations of such transforms with coefficients which are powers of $ t$.


References [Enhancements On Off] (What's this?)

  • [1] P. Appell, Sur les transformations des équations différentielles linéaires, C. R. Acad. Sci. Paris 91 (1880), 211-214.
  • [2] J. D'Archangelo, Linear ordinary differential equations with Laplace-Stieltjes transforms as coefficients, Trans. Amer. Math. Soc. 195 (1974), 115-145. MR 0344563 (49:9302)
  • [3] P. Hartman, On differential equations, Volterra equations and the functions $ J_\mu ^2 + Y_\mu ^2$, Amer. J. Math. 95 (1973), 553-593. MR 0333308 (48:11633)
  • [4] -, Ordinary differential equations, Baltimore, Md., 1973.
  • [5] G. N. Watson, A treatise on the theory of Bessel functions, 2nd ed., Cambridge Univ. Press, Cambridge; Macmillan, New York, 1944. MR 6, 64. MR 0010746 (6:64a)
  • [6] E. T. Whittaker, On a new connexion of Bessel functions with Legendre functions, Proc. London Math. Soc. 35 (1903), 198-206.
  • [7] A Wintner, On the existence of Laplace solutions for linear differential equations of second order, Amer. J. Math. 72 (1950), 442-450. MR 12, 179; 1002. MR 0036889 (12:179d)
  • [8] -, On the small divisors in integrations by Laplace transforms, Amer. J. Math. 73 (1951), 173-180. MR 13, 37. MR 0042003 (13:37b)
  • [9] -, Ordinary differential equations and Laplace transforms, Amer. J. Math. 79 (1957), 265-294. MR 19, 271. MR 0086940 (19:271b)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1975-0357935-8
Keywords: Laplace-Stieltjes transform, linear differential equations, Bessel and Legendre functions
Article copyright: © Copyright 1975 American Mathematical Society

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