By means of a change of
variable in the Riccati equation corresponding to the third order linear equation
Ly ≡ y′′′+ p(t)y′ + q(t)y = 0 a nonlinear integral equation is obtained which has a
solution obtainable by successive approximations under certain conditions on p and q.
This technique allows one to obtain new sharp comparison theorems for Ly = 0.
Several examples are given to illustrate the results.