Hammerstein integral equivalent of Riccati's equation

A transformation exists which allows the general Riccati equation\documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$$ \begin{array}{*{20}c}{{dy\left( r \right)} \mathord{\left/ {\vphantom {{dy\left( r \right)} {dr = A\left( r \right) + }}} \right. \kern-\nulldelimiterspace} {dr = A\left( r \right) + }}B\left( r \right)y\left( r \right) + C\left( r \right)y\left( r \right)^2 \hfill & 0\leqq r < b \end{array}$$\end{document} to be written in a simpler form:\documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$$ d\beta (r)/dr\, = \,P(r)\, + \,R(r)\beta (r)^2 \quad 0\buildrel{<}\over{=} r < b $$\end{document} The transformed equation has the equivalent nonlinear Hammerstein integral equation\documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$$ \begin{array}{*{20}c}\beta (r) = K\int_{r^{\prime} = 0}^b P(r^{\prime}) N(r, r^{\prime})dr^{\prime} \quad 0\buildrel{<}\over{=} r < b \end{array}$$\end{document} if the kernel N (r, r′) satisfies three conditions:\documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$$ \begin{array}{*{20}c} {({\rm i})} & {\{ d/dr - R(r)\beta (r)\} N(r,r)} \\ \end{array}\, = \,\delta (r,r)/K $$\end{document} and\documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$$ \begin{array}{*{20}c} {({\rm ii})} & {\{ d/dr'\, + \,R(r')\beta (r')\} N(r,r')} \\ \end{array}\, = \, - \delta (r,r')/K $$\end{document} and\documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$$ \begin{array}{*{20}c} {({\rm iii})} & {{\rm [}\beta (r')N(r,r'){\rm ]}_{r' = 0}^b } \\ \end{array} = 0 $$\end{document}A solution of the nonlinear integral equation is devised by repeatedly integrating the Hammerstein equation. During this procedure the kernel generates an equation that contains only coefficients of β( r ) 0 and β( r ) 1 . As a result, after truncating at the end of the n th cycle, it is a simple matter to write down a Padé‐type approximation: all coefficients in this approximation are capable of being evaluated in terms of simple algebraic formulations of P ( r ), R ( r ), and integrals over P ( r ). The zeroes of the denominator of the Padé‐type approximation define the points where singularities occur in β( r ).