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

282and thirdly, we get five more linear PDEs by picking the coefficients of cst.,of y 1 , of y 1,1 , of y 1 y 1,1 , of (y 1 ) 2 y 1,1 in (8.75)(8.83)⎧0 = Y x 1 x 1 x 1 + r Y x 1 x 1,⎪⎨⎪⎩0 = −3 Y x 1 x 1 y + X 1x 1 x 1 x + r Y 1 x 1 x 1 + r Y x 1 y + r X 1x 1 x 1,0 = Y x 1 y − Xx 1 1 x + r X 1 + r X 2 + r Y 1 x 1 + r Y y + r Xx 1 + r Y 1 x 1 x 10 = − 3 2 X 2x 1 x 1 + 3 Y yy − 9 X 1x 1 y + r X 1 + r X 2 + r X 1x 1 + r Y x 1 + r Y y++ r X 2x 1 + r X 1y + r Y x 1 x 1 + r Y x 1 y + r X 1x 1 x 1,0 = 6 X 1yy + 15 4 X 2x 1 y + r X 1 + r X 2 + r X 1x 1 + r Y x 1 + r Y y + r X 2x 1 + r X 1y + r X 2y ++ r Y x 1 x 1 + r Y x 1 y + r X 1x 1 x 1 + r X 2x 1 x 1 + r Y yy + r X 1x 1 y .Proposition 8.84. Setting as initial conditions the ten specific differentialcoefficients(8.85)P := P ( X 1 , X 2 , Y , Xy 1 , X 2x 2, Y x 1, Y y, X 2x 1 x 2, Y x 1 x 1, Y )yy= r X 1 + r X 2 + r Y + r Xy 1 + r X 2x + r Y 2 x 1 + r Y y + r X 2x 1 x + r Y 2 x 1 x 1 + r Y yy,it follows by cross differentiations and by linear substitutions from theLie equations (8.81) i , i = 1, 2, 3, 4, 5, (8.82) j , j = 1, 2, 3, (8.83) i , i =1, 2, 3, 4, 5, that X 1x, X 2 1 x, Y 1 x 2, X 1x, X 2 2 y , X 1x 1 y , X 2x 2 x, Y 2 x 1 x 2, X 1x 2 y ,X 2x 2 y , Y x 1 y, Xyy, 1 Y x 2 y, X 2x 1 x 1 x, Y 2 x 1 x 1 x 1, X 2x 1 x 2 xY 2 x 1 x 1 x 2, X 2x 1 x 2 y , Y x 1 x 1 y,Y x 1 yy, Y x 2 yy, Y yyy are uniquely determined as linear combinations of