Travaux sur les symétries de Lie des équations aux ... - DMA - Ens

+ [Y y 2 − 2 X xy ] y 1 y 2 + [−3 X y 2](y 1 ) 2 y 2 + [2 Y xy − X x 2] y 2 +7 8 6+ [Y y 2 − 2 X xy ] 2 y 1 y 2 + [−X y 2]3(y 1 ) 2 y 2 + [−3 X y ] (y 2 ) 2 +7 8 9+ [Y y − 2 X x ] y 3 + [−3 X y ] y 1 y 3 −10 11− [X x ] y 3 10− [X y ]y 1 y 3 . 11313We have underlined all the terms with a number appended. Each numberrefers to the order of appearance of the terms in the final simplified expressionof Y 3 , also written in [BK1989] with different notations:(2.8) ⎧Y 3 = Y x 3 + [3 Y x⎪⎨2 y − X x 3] y 1 + [3 Y xy 2 − 3 X x 2 y] (y 1 ) 2 ++ [Y y 3 − 3 X xy 2](y 1 ) 3 + [−X y 3](y 1 ) 4 + [3 Y xy − 3 X x 2]y 2 +⎪⎩+ [3 Y y 2 − 9 X xy ] y 1 y 2 + [−6 X y 2] (y 1 ) 2 y 2 + [−3 X y ] (y 2 ) 2 ++ [Y y − 3 X x ] y 3 + [−4 X y ]y 1 y 3 .After similar manual computations, the intermediate details of which wewill not copy in this Latex file, we get the desired expressions of Y 4 and ofY 5 . Firstly:(2.9)⎧Y 4 = Y x 4 + [ 4Y x 3 y − X x 4]y1 + [ 6Y x 2 y 2 − 4X x y] 3 (y1 ) 2 +⎪⎨⎪⎩+ [ 4Y xy 3 − 6X x 2 y 2 ](y1 ) 3 + [ Y y 4 − 4X xy 3](y1 ) 4 + [ −X y 4](y1 ) 5 ++ [ 6Y x 2 y − 4X x 3]y2 + [ 12Y xy 2 − 18X x 2 y]y1 y 2 ++ [ 6Y y 3 − 24X xy 2](y1 ) 2 y 2 + [ −10X y 3](y1 ) 3 y 2 ++ [ 3Y y 2 − 12X xy](y2 ) 2 + [ −15X y 2]y1 (y 2 ) 2 ++ [4Y xy − 6X x 2] y 3 + [ 4Y y 2 − 16X xy]y1 y 3 + [ −10X y 2](y1 ) 2 y 3 ++ [−10X y ]y 2 y 3 + [Y y − 4X x ] y 4 + [−5X y ] y 1 y 4 .