We prove the existence of Lp-solutions for a second order nonlinear Riccati-type scalar differential equation. This is applied to a perturbed third order linear differential equation with Lp- perturbations for some p ≥ 1. In this way, we know the asymptotic behavior of a solutiony and its derivatives showing error bounds. Furthermore, we obtain an asymptotic expansion in terms of Lp/i functions (1 ≤ i ≤ [p], [p] is the integer part of p), giving estimates for the error functions. The usefulness of these scalar results is shown studying the asymptotic behavior of a linear equation with unbounded coefficients. The asymptotic formulae obtained are simpler and more explicit than those obtained by 3 × 3 first order systems.