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Liouville-Green approximations via the Riccati transformation. (English) Zbl 0618.34011
Consider the scalar second order equation on (a,b) $u''=f(t)u$ with $f(t)=p(t)\sp 2+q(t)$ or $f(t)=-p(t)\sp 2+q(t),$ being p positive and piecewise differentiable and q piecewise continuous. Under suitable conditions on f, using a Riccati transformation, fundamental solutions are found and these are used to provide Liouville-Green approximations and explicit error bounds.
Reviewer: G.Caristi

##### MSC:
 34A45 Theoretical approximation of solutions of ODE 34B05 Linear boundary value problems for ODE
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##### References:
 [1] Atkinson, F. V.: The asymptotic solution of second order differential equations. Ann. mat. Pura appl. 37, 347-378 (1954) · Zbl 0056.08101 [2] Bellman, R.: Stability theory of differential equations. (1953) · Zbl 0053.24705 [3] Coppel, W. A.: Stability and asymptotic behavior of differential equations. (1965) · Zbl 0154.09301 [4] Coppel, W. A.: Dichotomies in stability theory. Lecture notes in mathematics 629 (1978) · Zbl 0376.34001 [5] Jr., W. A. Harris; Lutz, D. A.: On the asymptotic integration of linear differential systems. J. math. Anal. appl. 48, 1-16 (1974) · Zbl 0304.34043 [6] Jr., W. A. Harris; Lutz, D. A.: A unified theory of asymptotic integration. J. math. Anal. appl. 57, 571-586 (1977) · Zbl 0398.34012 [7] Hartman, P.; Wintner, A.: Asymptotic integrations of linear differential equations. Amer. J. Math. 77, 932 (1955) · Zbl 0064.08703 [8] Levinson, N.: The asymptotic nature of solutions of linear differential equations. Duke math. J. 15, 111-126 (1948) · Zbl 0040.19402 [9] Olver, F. W. J: Error bounds for the Liouville-Green (or WKB) approximation. Proc. Cambridge philos. Soc. 57, 790-810 (1961) · Zbl 0168.14003 [10] Olver, F. W. J: Asymptotics and special functions. (1974) · Zbl 0303.41035 [11] Perron, O.: Über stabilität und asymptotisches verhalten der integrale von differentialgleichungssystemen. Math. Z. 29, 129-160 (1929) · Zbl 54.0456.04 [12] Perron, O.: Über ein vermeintliches stabilitätskriterium. Math.-physik. Kl. fachgruppe 1 (1930) · Zbl 56.0380.03 [13] Smith, D. R.: Decoupling and order reduction via the Riccati transformation. (1984) [14] Smith, D. R.: Singular perturbation theory. (1985) · Zbl 0567.34055 [15] Taylor, J. G.: Error bounds for the Liouville-Green approximation to initial-value problems. Z. angew. Math. mech. 58, 529-537 (1978) · Zbl 0399.34004 [16] Taylor, J. G.: Improved error bounds for the Liouville-Green (or WKB) approximation. J. math. Anal. appl. 85, 79-89 (1982) · Zbl 0499.34035