2.20.3.5.3 Example \(y^{\prime }=3a-a^{2}x^{2}+y^{2}\)

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

\begin{align*} f_{0} & =3a-a^{2}x^{2}\\ f_{1} & =0\\ f_{2} & =1 \end{align*}

A particular solution is \(y_{1}=ax-\frac {1}{x}\,\). Using the direct formula (1) given earlier

\begin{align} y & =y_{1}+\Phi \frac {1}{c_{1}-\int \Phi f_{2}dx}\nonumber \\ & =ax-\frac {1}{x}+\frac {\Phi }{c_{1}-\int \Phi dx} \tag {B}\end{align}

Where

\begin{align*} \Phi & =e^{\int 2f_{2}y_{1}+f_{1}dx}\\ & =e^{\int 2\left ( ax-\frac {1}{x}\right ) dx}\\ & =e^{ax^{2}-2\ln x}\\ & =\frac {e^{ax^{2}}}{x^{2}}\end{align*}

Hence (B) becomes

\begin{align*} y & =ax-\frac {1}{x}+\frac {\frac {e^{ax^{2}}}{x^{2}}}{c_{1}-\int \frac {e^{ax^{2}}}{x^{2}}dx}\\ & =ax-\frac {1}{x}+\frac {e^{ax^{2}}}{x^{2}\left ( c_{1}-\int \frac {e^{ax^{2}}}{x^{2}}dx\right ) }\end{align*}