1.3.6.3 Example when G is zero \(y^{\prime }=3+x^{2}-\frac {2}{x}y-x^{2}y^{2}\)

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

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

Hence

\begin{align*} F & =f_{0}f_{2}=\left ( 3+x^{2}\right ) \left ( -x^{2}\right ) \\ & =-3x^{2}-x^{4}\\ G & =f_{1}+\frac {f_{2}^{\prime }}{f_{2}}=-\frac {2}{x}+\frac {-2x}{-x^{2}}=-\frac {2}{x}+\frac {2}{x}\\ & =0 \end{align*}

The new ode is (after substituting \(y=\frac {u}{f_{2}}\)) becomes

\begin{align} u^{\prime } & =F\left ( x\right ) +G\left ( x\right ) u+u^{2}\nonumber \\ & =\left ( -3x^{2}-x^{4}\right ) +u^{2} \tag {1}\end{align}

Now we look at the degree of \(F\left ( x\right ) \). Since degree is even, then polynomial solution might exist for the above ode. Now we expand \(\sqrt {-F\left ( x\right ) }\) in series and stop at the constant term. The degree of \(F\left ( x\right ) \) is \(4\). Hence \(2n=4\) and \(n=2\).

\begin{align*} \sqrt {-F\left ( x\right ) } & =\sqrt {3x^{2}+x^{4}}\\ X\left ( x\right ) & =\overbrace {a_{2}x^{2}+a_{1}x+a_{0}}+\frac {b_{1}}{x}+\frac {b_{2}}{x^{2}}+\cdots \end{align*}

To find \(X\left ( x\right ) \), we use method of undetermined coefficients. Let

\begin{align*} \left ( a_{2}x^{2}+a_{1}x+a_{0}\right ) ^{2} & =3x^{2}+x^{4}\\ \left ( a_{2}x^{2}+\left ( a_{1}x+a_{0}\right ) \right ) ^{2} & =3x^{2}+x^{4}\\ a_{2}x^{4}+\left ( a_{1}x+a_{0}\right ) ^{2}+2a_{2}x^{2}\left ( a_{1}x+a_{0}\right ) & =3x^{2}+x^{4}\\ a_{2}x^{4}+\left ( a_{1}^{2}x^{2}+a_{0}^{2}+2a_{0}a_{1}x\right ) +2a_{1}a_{2}x^{3}+2a_{0}a_{2}x^{2} & =3x^{2}+x^{4}\\ x^{4}\left ( a_{2}\right ) +x^{3}\left ( 2a_{1}a_{2}\right ) +x^{2}\left ( a_{1}^{2}+2a_{0}a_{2}\right ) +x\left ( 2a_{0}a_{1}\right ) +a_{0}^{2} & =3x^{2}+x^{4}\end{align*}

Hence \(a_{2}=1,2a_{1}a_{2}=0\) or \(a_{1}=0\) and \(a_{1}^{2}+2a_{0}a_{2}=3\) or \(2a_{0}=3\) or \(a_{0}=\frac {3}{2}\). Hence

\[ X\left ( x\right ) =x^{2}+\frac {3}{2}\]

Now we test if \(\pm X\left ( x\right ) \) satisfies (1). It does not. Hence no polynomial solution.