Clairautの方程式

以下の2つ形式の微分方程式のことを示す言葉

$ u=xu'+f\left(u'\right)

$ u=x\frac{\partial u}{\partial y}+y\frac{\partial u}{\partial x}+f\left(\frac{\partial u}{\partial x},\frac{\partial u}{\partial y}\right)

解法

$ u=xu'+f\left(u'\right)

$ \iff\begin{dcases}u&=xu'+f\left(u'\right)\\u'&=u'+xu''+f'\left(u'\right)u''\end{dcases}

$ \iff\begin{dcases}u&=xu'+f\left(u'\right)\\0&=u''\left(x+f'\left(u'\right)\right)\end{dcases}

$ \iff\begin{dcases}u=xu'+f\left(u'\right)\\\exist C\in\Complex:u'=C\lor x+f'(u')=0\end{dcases}

$ \iff \exist C\in\Complex:u=Cx+f(C)\lor\begin{dcases}u=xu'+f\left(u'\right)\\x+f'(u')=0\end{dcases}

$ \iff \exist C\in\Complex:u=Cx+f(C)\lor u=x{f'}^{-1}(-x)+f\circ{f'}^{-1}(-x)