Find the equation of the parabola that satisfies the following conditions: Vertex (0, 0),

Since the vertex is $(0,0)$ and the parabola is symmetric about the $y$-axis, the equation of the parabola is either of the form $x^{2}=4 a y$ or $x^{2}=-4 a y$.

The parabola passes through point (5, 2), which lies in the first quadrant.

Therefore, the equation of the parabola is of the form $x^{2}=4 a y$, while point

$(5,2)$ must satisfy the equation $x^{2}=4 a y$.

$\therefore(5)^{2}=4 \times a \times 2 \Rightarrow 25=8 a \Rightarrow a=\frac{25}{8}$

Thus, the equation of the parabola is

$x^{2}=4\left(\frac{25}{8}\right) y$

$2 x^{2}=25 y$