On bounds for monotonic first order differential systems and the Liouville�Green approximation

• Autores: José Javier Segura Sala
• Localización: Journal of approximation theory, ISSN 0021-9045, Vol. 170, Nº 1, 2013, págs. 107-115
• Idioma: inglés
• DOI: 10.1016/j.jat.2012.07.003
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  • Many special functions are solutions of first order linear systems y��(x) = a(x)y(x) + d(x)w(x), w��(x) = b(x)w(x) + e(x)y(x). We obtain bounds for the logarithmic derivatives of the solutions of monotonic systems satisfying certain initial conditions. In particular, by Liouville-transforming the first order system equivalent to the second order ODE y����(x) + A(x)y(x) = 0 we obtain bounds related to the Liouville.Green approximation and find conditions under which such approximation is a bound for some solutions. We illustrate this with the Airy equation.