Differential Equations

$\left|\begin{array}{ccc}x^2 & y^2 & x+y \\ 2 x & 2 y^{\prime} y & y^{\prime}+1 \\ 2 & 2 y y^{\prime \prime} & y^{\prime \prime}\end{array}\right|=0$

$\left|\begin{array}{ccc}x^2 & y^2 & x+y \\ 2 x & 2 y y^{\prime} & 1+y^{\prime} \\ 2 & 2\left(y^{\prime 2}+y y^{\prime \prime}\right) & y^{\prime \prime}\end{array}\right|=0$

$\left|\begin{array}{ccc}x^2 & y^2 & x+y \\ 2 x & 2 y y^{\prime} & y+1 \\ 2 & 2\left(y^{\prime 2}+y^{\prime} y^{\prime \prime}\right) & y^{\prime \prime}\end{array}\right|=0$

$\left|\begin{array}{ccc}x^2 & y^2 & x+y \\ 2 x & 2 y & 1+y^{\prime} \\ 2 & 2 y y^{\prime} y^{\prime \prime} & y^{\prime \prime}\end{array}\right|=0$