# System of differential equations

I've tried to solve the following simple system:

$\Bigg\{ \begin{array}{cc} \dot{x}=y \\ \dot{y}=\frac{y^2}{x} \end{array}$

After some algebra (differentiating the first equation, then plugging in the square of $\dot{x}$, then differentiating the second expression, plugging in the expression for $\dot{x}$ from the first and for $x$ from $x=\frac{y^2}{\dot{y}}$. I got the following two equations for each variable that made me think something went wrong somewhere:

$\Bigg\{ \begin{array}{cc} \ddot{x} x -\dot{x}^2=0 \\ \ddot{y} y -\dot{y}^2=0 \end{array}$

I probably made some silly mistake, in such case pls don't solve it for me, just point in the right direction. Thanks.

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Hint: Compute the derivative of $\dfrac{\dot{x}}x$.