Solutions for certain classes of the Riccati differential equation

Saad N., Hall R. L., Ciftci H.

JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, vol.40, no.35, pp.10903-10914, 2007 (SCI-Expanded) identifier identifier


We derive some analytic closed-form solutions for a class of Riccati equations y'(x) - lambda(0)(x) y(x) +/- y(2)(x) = +/- s(0)(x) where lambda(0)( x), s(0)( x) are C-infinity- functions. We show that if delta(n) = lambda(n)s(n-1) - lambda(n-1)s(n) = 0, where lambda(n) = lambda(')(n-1) + s(n-1) + lambda(0)lambda(n-1) and s(n) = s(n-1)(')+ s(0)lambda(k-1) , n = 1, 2, ... , then the Riccati equation has a solution given by y( x) = -/+ s(n-1)(x)/ lambda(n-1)(x). Extension to the generalized Riccati equation y'(x) + P(x) y(x) + Q(x) y(2)(x) = R(x) is also investigated.