1.3.5.4 Example \(y^{\prime }=ax^{n-1}+ax^{n}y+y^{2}\)

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

\begin{align*} f_{0} & =ax^{n-1}\\ f_{1} & =ax^{n}\\ f_{2} & =1 \end{align*}

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

\begin{align} y & =y_{1}+\Phi \frac {1}{c_{1}-\int \Phi f_{2}dx}\nonumber \\ & =-\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 -\frac {2}{x}+ax^{n}dx}\\ & =e^{a\frac {x^{n+1}}{n+1}-2\ln x}\\ & =\frac {1}{x^{2}}e^{a\frac {x^{n+1}}{n+1}}\end{align*}

Hence (B) becomes

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