ダランベールの微分方程式

${\displaystyle y=xf(p)+g(p)\,}$　　…　(2)

${\displaystyle p=f(p)+xf'(p){\frac {dp}{dx}}+g'(p){\frac {dp}{dx}}}$

${\displaystyle {\frac {dx}{dp}}+{\frac {f'(p)}{f(p)-p}}x+{\frac {g'(p)}{f(p)-p}}=0}$　　…　(3)

${\displaystyle {\begin{aligned}x=&\exp \left(-\int {\frac {f'(p)}{f(p)-p}}dp\right)\\&\left\{C-\int {\frac {g'(p)}{f(p)-p}}\exp \left(\int {\frac {f'(p)}{f(p)-p}}dp\right)dp\right\}\\\end{aligned}}}$　　…　(4)