This section shows how to obtain eq. (8) in paper "Computer Algebra Solving of First Order
ODEs Using Symmetry Methods" 1996 by Durate, Terrab, Mota. Which is an alternative
equation to solve instead of the main Lie condition for symmetry we were looking at
above.
Starting with the main linearized symmetry pde
\begin{equation} \eta _{x}+\omega \left ( \eta _{y}-\xi _{x}\right ) -\omega ^{2}\xi _{y}-\omega _{x}\xi -\omega _{y}\eta =0 \tag {14}\end{equation}
Assuming ansatz
\begin{equation} \eta =\xi \omega +\chi \tag {A}\end{equation}
Hence
\begin{align*} \eta _{x} & =\xi _{x}\omega +\xi \omega _{x}+\chi _{x}\\ \eta _{y} & =\xi _{y}\omega +\xi \omega _{y}+\chi _{y}\end{align*}
Then (14) becomes
\begin{align} \left ( \xi _{x}\omega +\xi \omega _{x}+\chi _{x}\right ) +\omega \left ( \left ( \xi _{y}\omega +\xi \omega _{y}+\chi _{y}\right ) -\xi _{x}\right ) -\omega ^{2}\xi _{y}-\omega _{x}\xi -\omega _{y}\left ( \xi \omega +\chi \right ) & =0\nonumber \\ \xi _{x}\omega +\xi \omega _{x}+\chi _{x}+\xi _{y}\omega ^{2}+\xi \omega _{y}\omega +\chi _{y}\omega -\omega \xi _{x}-\omega ^{2}\xi _{y}-\omega _{x}\xi -\xi \omega \omega _{y}-\omega _{y}\chi & =0\nonumber \\ \xi _{x}\omega +\chi _{x}+\xi _{y}\omega ^{2}+\xi \omega _{y}\omega +\chi _{y}\omega -\omega \xi _{x}-\omega ^{2}\xi _{y}-\xi \omega \omega _{y}-\omega _{y}\chi & =0\nonumber \\ \chi _{x}+\xi _{y}\omega ^{2}+\xi \omega _{y}\omega +\chi _{y}\omega -\omega ^{2}\xi _{y}-\xi \omega \omega _{y}-\omega _{y}\chi & =0\nonumber \\ \chi _{x}+\xi \omega _{y}\omega +\chi _{y}\omega -\xi \omega \omega _{y}-\omega _{y}\chi & =0\nonumber \end{align}
Or
\begin{equation} \chi _{x}+\chi _{y}\omega -\omega _{y}\chi =0 \tag {1}\end{equation}
And hence (1) is now solved for
\(\chi \left ( x,y\right ) \). If we are able to find
\(\chi \) then we can use the ansatz
\(\eta =\xi \omega +\chi \). This leaves
only one unknown
\(\xi \). The paper does not explain how to solve for this,
\(\xi \), which I assume is by using
(14) again. The paper only said
The knowledge of \(\chi \), in turn, allows one to set \(\xi \) and \(\eta ~\)as desired using (A)
Which is not too clear how in practice this is done. I need to work an example showing this. The
paper says that (1) is solved for \(\chi \left ( x,y\right ) \) by using bivariate polynomial ansatz. The degree can be set by a
user, or Maple internally determines this.
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