Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 2.djvu/208

l’équation différentielle précédente donnera, en l’intégrant,

${\rm {Y}}^{(i)}=l{\rm {M}}^{(i)}\lambda _{i}t+l{\rm {N}}^{(i)}\cos \lambda _{i}t$

$+{\frac {i(i+1)}{\lambda _{i}}}l\sin \lambda _{i}t.\int {\rm {U}}^{(i)}dt\cos \lambda _{i}t-{\frac {i(i+1)}{\lambda _{i}}}l\cos \lambda _{i}t.\int {\rm {U}}^{(i)}dt\sin \lambda _{i}t,$

${\rm {M}}^{(i)}$ et ${\rm {N}}^{(i)}$ étant des fonctions rationnelles et entières de $\mu ,{\sqrt {1-\mu ^{2}}}\sin \varpi$ et ${\sqrt {1-\mu ^{2}}}\cos \varpi ,$ qui satisfont aux équations à différences partielles

${\begin{aligned}0&={\frac {\partial .\left(1-\mu ^{2}\right){\frac {\partial {\rm {M}}^{(i)}}{\partial \mu }}}{\partial \mu }}+{\frac {\frac {\partial ^{2}{\rm {M}}^{(i)}}{\partial \varpi ^{2}}}{1-\mu ^{2}}}+i(i+1){\rm {M}}^{(i)},\\\\0&={\frac {\partial .\left(1-\mu ^{2}\right){\frac {\partial {\rm {N}}^{(i)}}{\partial \mu }}}{\partial \mu }}+{\frac {\frac {\partial ^{2}{\rm {N}}^{(i)}}{\partial \varpi ^{2}}}{1-\mu ^{2}}}+i(i+1){\rm {N}}^{(i)},\end{aligned}}$

L’équation différentielle en ${\rm {Y}}^{(t)}$ donne, en y supposant $i=0,\ {\frac {\partial ^{2}{\rm {Y^{(0)^{2}}}}}{\partial t}}=0,$ et par conséquent

${\rm {Y}}^{(0)}=l{\rm {M}}^{(0)}t+l{\rm {N}}^{(0)}\,;$

l’équation

$y={\rm {Y^{(0)}+Y^{(1)}+Y^{(2)}+\ldots }}$

donnera donc

${\begin{aligned}y=l{\rm {M}}^{(0)}t+l{\rm {N}}^{(0)}&+l{\rm {M}}^{(1)}\sin \lambda _{1}t+l{\rm {N}}^{(1)}\lambda _{1}t\\&+l{\rm {M}}^{(2)}\sin \lambda _{2}t+l{\rm {N}}^{(2)}\lambda _{2}t\\&+\ldots \ldots \ldots \ldots \ldots \ldots \ldots \\&+l{\rm {M}}^{(i)}\sin \lambda _{i}t+l{\rm {N}}^{(i)}\lambda _{i}t\\&+\ldots \ldots \ldots \ldots \ldots \ldots \ldots \\&+{\frac {6l}{\lambda _{2}}}\sin \lambda _{2}t.\int {\rm {U}}^{(2)}dt\cos \lambda _{2}t\\&-{\frac {6l}{\lambda _{2}}}\cos \lambda _{2}t.\int {\rm {U}}^{(2)}dt\sin \lambda _{2}t\\&+\ldots \ldots \ldots \ldots \ldots \ldots \ldots \\&+{\frac {i(i+1)l}{\lambda _{i}}}\sin \lambda _{i}t.\int {\rm {U}}^{(i)}dt\cos \lambda _{i}t\\&-{\frac {i(i+1)l}{\lambda _{i}}}\cos \lambda _{i}t.\int {\rm {U}}^{(i)}dt\sin \lambda _{i}t\\&+\ldots \ldots \ldots \ldots \ldots \ldots \ldots \end{aligned}}$