Abstract
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.
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References
Bellman, R.,Perturbation Techniques in Mathematics, Physics, and Engineering, Holt, Rinehart, and Winston, New York, New York, 1964.
Courant, R.,Differential and Integral Calculus, Vol. 2, Interscience Publishers, New York, New York, 1961.
Ford, L. R.,Differential Equations, McGraw-Hill, New York, New York, 1955.
Ince, E. L.,Ordinary Differential Equations, Dover, New York, New York, 1956.
Murphy, G. M.,Ordinary Differential Equations and Their Solutions, D. Van Nostrand, New York, New York, 1960.
Roberts, S. M., andShipman, J. S.,Two-Point Boundary-Value Problems: Shooting Methods, American Elsevier, New York, New York, 1972.
Roberts, S. M., andShipman, J. S.,A New Derivation for the Scott Formulation of Invariant Imbedding, Journal of Mathematical Analysis and Applications, Vol. 64, pp. 480–487, 1978.
Roberts, S. M.,Direct Use of the Fundamental Matrix in Two-Point Boundary-Value Problems, Journal of Mathematical Analysis and Applications, Vol. 71, pp. 147–160, 1979.
Roberts, S. M., andShipman, J. S.,On the Formulation of Invariant Imbedding Problems, Journal of Optimization Theory and Applications, Vol. 28, pp. 525–547, 1979.
Scott, M. R.,Invariant Imbedding and Its Applications to Ordinary Differential Equations, Addison-Wesley, Reading, Massachusetts, 1973.
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Roberts, S.M. Boundary conditions of the second-order differential equation and the Riccati equation. J Optim Theory Appl 40, 397–403 (1983). https://doi.org/10.1007/BF00933507
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DOI: https://doi.org/10.1007/BF00933507