Page:Annales de mathématiques pures et appliquées, 1820-1821, Tome 11.djvu/116

108

INTÉGRATION

${\begin{aligned}T&=2\left(S{\frac {\operatorname {d} V}{\operatorname {d} x}}-V{\frac {\operatorname {d} S}{\operatorname {d} x}}\right)-\quad S(R+R')=-4x(2+x)\left(1-x^{2}\right),\\\\T'&=2\left(S'{\frac {\operatorname {d} V}{\operatorname {d} x}}-V{\frac {\operatorname {d} S'}{\operatorname {d} x}}\right)-S'(R+R')=+4x(2-x)\left(1-x^{2}\right),\\\\U&=2V\left(R{\frac {\operatorname {d} S}{\operatorname {d} x}}-S{\frac {\operatorname {d} R}{\operatorname {d} x}}\right)+\quad \ S(RR'-SS')=8x^{2}\left(1-x^{2}\right)^{2},\\\\U'&=2V\left(R'{\frac {\operatorname {d} S'}{\operatorname {d} x}}-S'{\frac {\operatorname {d} R'}{\operatorname {d} x}}\right)+S'(RR'-SS')=8x^{2}\left(1-x^{2}\right)^{2}\,;\end{aligned}}$

substituant donc dans nos équations du second ordre, en $p$ et $q,$ elles deviendront

${\begin{aligned}4x^{2}\left(1-x^{2}\right)^{2}\left(1+4x+x^{2}\right){\frac {\operatorname {d} ^{2}p}{\operatorname {d} x^{2}}}-8x^{2}\left(1-x^{2}\right)^{2}(2+x){\frac {\operatorname {d} p}{\operatorname {d} x}}+8x^{2}\left(1-x^{2}\right)^{2}p&=0,\\\\4x^{2}\left(1-x^{2}\right)^{2}\left(1-4x+x^{2}\right){\frac {\operatorname {d} ^{2}q}{\operatorname {d} x^{2}}}+8x^{2}\left(1-x^{2}\right)^{2}(2-x){\frac {\operatorname {d} q}{\operatorname {d} x}}+8x^{2}\left(1-x^{2}\right)^{2}q&=0,\end{aligned}}$

ou, en simplifiant,

${\begin{aligned}\left(1+4x+x^{2}\right){\frac {\operatorname {d} ^{2}p}{\operatorname {d} x^{2}}}-2(2+x){\frac {\operatorname {d} p}{\operatorname {d} x}}+2p&=0,\\\\\left(1-4x+x^{2}\right){\frac {\operatorname {d} ^{2}q}{\operatorname {d} x^{2}}}+2(2-x){\frac {\operatorname {d} q}{\operatorname {d} x}}+2q&=0.\end{aligned}}$

Les équations qui devront donner $p'$ et $q'$ seront donc

${\begin{aligned}\left(1+4x+x^{2}\right){\frac {\operatorname {d} ^{2}p'}{\operatorname {d} x^{2}}}-2(2+x){\frac {\operatorname {d} p'}{\operatorname {d} x}}+2p'&=0,\\\\\left(1-4x+x^{2}\right){\frac {\operatorname {d} ^{2}q'}{\operatorname {d} x^{2}}}+2(2-x){\frac {\operatorname {d} q'}{\operatorname {d} x}}+2q'&=0.\end{aligned}}$