A parabola is a set of all points in a plane that are equidistant from a given fixed point (the Focus) and a given straight line (the Directrix).

Algebraic equation
$x^2 - 4ya = 0$

Cartesian equation
$y = \frac{x^{2}}{4a}$

Parametric equations
$x(t) = 2 t a$
$y(t) = t^2 a$

Polar equation
$r(\theta) = 4 tan(\theta) sec(\theta) a$

Differential equation
$3y'(x) y'' (x^{2})-y^{3}(x)y'(x^{2}) =0$