1.12.22 Example \(y^{\prime }=\frac {y-xf\left ( x^{2}+ay^{2}\right ) }{x+ayf\left ( x^{2}+ay^{2}\right ) }\)

\begin{align*} y^{\prime } & =\frac {y-xf\left ( x^{2}+ay^{2}\right ) }{x+ayf\left ( x^{2}+ay^{2}\right ) }\\ & =\omega \left ( x,y\right ) \end{align*}

Using ansatz it is found that

\begin{align*} \xi & =-ay\\ \eta & =x \end{align*}

Hence

\begin{align} \frac {dx}{\xi } & =\frac {dy}{\eta }=dS\nonumber \\ \frac {dx}{-ay} & =\frac {dy}{x}=dS \tag {1}\end{align}

The first two give

\[ \frac {dy}{dx}=\frac {x}{-ay}\]

This is separable. Solving gives (taking one root)

\[ y=\frac {\sqrt {a\left ( ac_{1}-x^{2}\right ) }}{a}\]

Solving for \(c_{1}\) gives

\[ c_{1}=\frac {x^{2}+ay^{2}}{a}\]

Hence

\[ X=\frac {x^{2}+ay^{2}}{a}\]

\(Y\) is found from either \(\frac {dy}{\eta }=dY\,\) or \(\frac {dx}{\xi }=dY\). Using \(\frac {dx}{-ay}=dY\) then

\[ \frac {dx}{-ay}=dY \]

But \(y=\frac {\sqrt {a\left ( ac_{1}-x^{2}\right ) }}{a}\). Hence

\begin{align*} \frac {dx}{-a\frac {\sqrt {a\left ( ac_{1}-x^{2}\right ) }}{a}} & =dY\\ \frac {dx}{-\sqrt {a\left ( ac_{1}-x^{2}\right ) }} & =dY\\ -\frac {1}{\sqrt {a}}\arctan \left ( \frac {\sqrt {a}x}{\sqrt {c_{1}a^{2}-x^{2}a}}\right ) & =Y\\ -\frac {1}{\sqrt {a}}\arctan \left ( \frac {\sqrt {a}x}{ay}\right ) & =Y \end{align*}

Where constant of integration is set to zero. What is left is to find \(\frac {dY}{dX}\). This is given by

\begin{equation} \frac {dY}{dX}=\frac {Y_{x}+Y_{y}\omega }{X_{x}+X_{y}\omega } \tag {2}\end{equation}

But

\begin{align*} X_{x} & =\frac {2x}{a}\\ X_{y} & =2y\\ Y_{x} & =-\frac {y}{x^{2}y^{2}+a}\\ Y_{y} & =-\frac {x}{a\left ( 1+\frac {x^{2}y^{2}}{a}\right ) }\end{align*}

Hence (2) becomes

\[ \frac {dY}{dX}=\frac {-\frac {y}{x^{2}y^{2}+a}+\left ( -\frac {x}{a\left ( 1+\frac {x^{2}y^{2}}{a}\right ) }\right ) \omega }{\frac {2x}{a}+2y\omega }\]

But \(X=\frac {x^{2}+ay^{2}}{a}\). The above becomes

\[ \frac {dY}{dX}=\frac {-\frac {y}{aX}+\left ( -\frac {x}{a\left ( 1+\frac {x^{2}y^{2}}{a}\right ) }\right ) \omega }{\frac {2x}{a}+2y\omega }\]

To finish. Another hard part of this Lie method is to convert back \(\,\frac {dS}{dR}=\frac {S_{x}+S_{y}\omega }{R_{x}+R_{y}\omega }\) so that the RHS is only a function of \(R\). Need to find a robust way to do this. This is now a weak point in my program as I have few ode’s that it can’t do it