Page:Annales de mathématiques pures et appliquées, 1826-1827, Tome 17.djvu/78

$e^{\int t\operatorname {d} x}\quad$ou$\quad y={\frac {e^{\int {\frac {Q\operatorname {d} x}{y}}}}{e^{\int P\operatorname {d} x}}}=e^{-\int P\operatorname {d} x}.e^{\int {\frac {Q\operatorname {d} x}{y}}}.\qquad$(1)

Or, on a

$\operatorname {d} .e^{\int {\frac {Q\operatorname {d} x}{y}}}={\frac {Q\operatorname {d} x}{y}}.e^{\int {\frac {Q\operatorname {d} x}{y}}},$

et, en intégrant

$e^{\int {\frac {Q\operatorname {d} x}{y}}}=\int {\frac {Q}{y}}.e^{\int {\frac {Q\operatorname {d} x}{y}}}\operatorname {d} x.\qquad$(2)

Mais, l’équation (1) donne

$e^{\int {\frac {Q\operatorname {d} x}{y}}}=ye^{\int P\operatorname {d} x},$

mettant donc cette valeur dans l’équation (2), elle deviendra

$ye^{\int P\operatorname {d} x}=\int {\frac {Q}{y}}.ye^{\int P\operatorname {d} x}.\operatorname {d} x,$

ou, en simplifiant et tirant ensuite la valeur de $y$

$y=e^{-\int P\operatorname {d} x}.\int e^{\int P\operatorname {d} x}.Q\operatorname {d} x,$

formule cherchée.