Roberts, S.M. J Optim Theory Appl (1983) 40: 397. doi:10.1007/BF00933507
The conversion of a second-order linear ordinary differential equation with variable coefficients into a Riccati equation depends on whether the second-order problem is an initial-value or two-point boundary-value problem. The distinction is critical in determining the initial condition for the Riccati equation. If the second-order problem is an initial-value problem, the choice of the Riccati transformation depends on whether a zero initial condition for the function or its derivative is specified. If the problem is a two-point boundary-value problem, special methods must be introduced as described in the paper.
Riccati equation second-order ordinary differential equation initial-value problem two-point boundary-value problem method of adjoints invariant imbedding Scott's method