I need opinions on how to solve these equations. I looked into Laplace-transform but have not tried anything yet. I want help before proceeding.
$\frac{dx}{dt}=-2\sqrt{x^2+y^2}\cdot x^2$
$\frac{dy}{dt}=-10-2\sqrt{x^2+y^2}\cdot y^2$
Thanks in advance.
The drag or friction in air is indeed of size $cv^2$, but its direction is $-\hat v$ where $\hat v=\frac{\vec v}{v}$, so that in total the vector force is $-cv\vec v$ (using $v=\|\vec v\|$). This gives the component equations \begin{align} \frac{dx}{dt}&=-2\sqrt{x^2+y^2}\cdot x \\ \frac{dy}{dt}&=-10-2\sqrt{x^2+y^2}\cdot y \end{align} There is indeed a first integral to be found, it has a whole wunderkind story attached (more a re-discovery of a decades old observation), but as this first integral also only gives an implicit non-linear equation between $x$ and $y$ its practical usefulness is in doubt. Use numerical methods.