2.20.3.5.5 Example \(xy^{\prime }=-ab^{2}x^{n+2m}+my+ax^{n}y^{2}\)

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

\begin{align*} f_{0} & =-ab^{2}x^{n+2m}\\ f_{1} & =m\\ f_{2} & =ax^{n}\\ g & =x \end{align*}

A particular solution is \(y_{1}=bx^{m}\,\). Using the direct formula (1) given earlier

\begin{align} y & =y_{1}+\Phi \frac {1}{c_{1}-\int \Phi \frac {f_{2}}{g}dx}\nonumber \\ \Phi & =e^{\int \frac {2f_{2}y_{1}+f_{1}}{g}dx} \tag {2}\end{align}

Then

\begin{align*} \Phi & =e^{\int \frac {2\left ( ax^{n}\right ) \left ( bx^{m}\right ) +m}{x}dx}\\ & =x^{m}e^{\frac {2abx^{n+m}}{n+m}}\end{align*}

Hence the solution is

\begin{align*} y & =y_{1}+\Phi \frac {1}{c_{1}-\int \Phi \frac {f_{2}}{g}dx}\\ & =bx^{m}+\frac {x^{m}e^{\frac {2abx^{n+m}}{n+m}}}{c_{1}-\int x^{m}e^{\frac {2abx^{n+m}}{n+m}}\frac {\left ( ax^{n}\right ) }{x}dx}\\ & =bx^{m}+\frac {x^{m}e^{\frac {2abx^{n+m}}{n+m}}}{c_{1}-a\int x^{m+n-1}e^{\frac {2abx^{n+m}}{n+m}}dx}\\ & =bx^{m}+\frac {x^{m}e^{\frac {2abx^{n+m}}{n+m}}}{c_{1}-a\left ( \frac {e^{\frac {2abx^{n+m}}{n+m}}}{2ab}\right ) }\\ & =bx^{m}+\frac {2bx^{m}e^{\frac {2abx^{n+m}}{n+m}}}{2bc_{1}-e^{\frac {2abx^{n+m}}{n+m}}}\\ & =bx^{m}+\frac {2bx^{m}\Delta }{2bc_{1}-\Delta }\end{align*}

Where

\[ \Delta =e^{\frac {2abx^{n+m}}{n+m}}\]