Boundary conditions of the second-order differential equation and the Riccati equation
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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.
Key WordsRiccati equation second-order ordinary differential equation initial-value problem two-point boundary-value problem method of adjoints invariant imbedding Scott's method
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