Integration of ordinary linear differential equations by Laplace-Stieltjes transforms

D’Archangelo, James; Hartman, Philip

doi:10.1090/S0002-9947-1975-0357935-8

Integration of ordinary linear differential equations by Laplace-Stieltjes transforms
HTML articles powered by AMS MathViewer

by James D’Archangelo and Philip Hartman PDF

Trans. Amer. Math. Soc. 204 (1975), 245-266 Request permission

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

Sur les transformations des équations différentielles linéaires

91

James D’Archangelo, Linear ordinary differential equations with Laplace-Stieltjes transforms as coefficients, Trans. Amer. Math. Soc. 195 (1974), 115–145. MR 344563, DOI 10.1090/S0002-9947-1974-0344563-2
Philip Hartman, On differential equations, Volterra equations and the function $J_{\mu }^{2}+Y_{\mu }^{2}$, Amer. J. Math. 95 (1973), 553–593. MR 333308, DOI 10.2307/2373730

Ordinary differential equations

G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR 0010746

On a new connexion of Bessel functions with Legendre functions

35

Aurel Wintner, On the existence of Laplace solutions for linear differential equations of second order, Amer. J. Math. 72 (1950), 442–450. MR 36889, DOI 10.2307/2372047
Aurel Wintner, On the small divisors in integrations by Laplace transforms, Amer. J. Math. 73 (1951), 173–180. MR 42003, DOI 10.2307/2372168
Aurel Wintner, Ordinary differential equations and Laplace transforms, Amer. J. Math. 79 (1957), 265–294. MR 86940, DOI 10.2307/2372682