Write down the Lagrange-Charpit equations for the equation

$\displaystyle

\displaystyle \frac{\partial u}{\partial x}\frac{\partial u}{\partial y}-y\frac{\partial u}{\partial x}-x

\frac{\partial u}{\partial y}=0

$ and use them to show that

$\displaystyle

\displaystyle \frac{d^2 p}{dt^2}=p, \frac{d^2 q}{dt^2}=q,\frac{d^2 x}{dt^2}=x,\frac{d^2 y}{dt^2}=y

$

For the first term, I attempted the folllowing

$\displaystyle

\displaystyle \frac{d}{dt}\frac{dp}{dt}=\frac{d}{dt}(-F_x-pF_u)=\frac{d}{dt}(-(-q))=\frac{dq}{dt}

$

Cannot see any link from this to continue on....

Thanks