1.3.6.8 Example when G,F not polynomials \(y^{\prime }=\frac {7}{x}-\frac {7}{x}y-y^{2}\)

Comparing to \(y^{\prime }=f_{0}+f_{1}y+f_{2}y^{2}\)

\begin{align*} f_{0} & =\frac {7}{x}\\ f_{1} & =-\frac {7}{x}\\ f_{2} & =-1 \end{align*}

Hence

\begin{align*} F & =f_{0}f_{2}=-\frac {7}{x}\\ G & =f_{1}+\frac {f_{2}^{\prime }}{f_{2}}=-\frac {7}{x}\end{align*}

\(G,F\) are not polynomials. Hence can not use this section method. This however can be solved by transforming to second order ode.