A family of parabolas $y=ax^{2}$ $(a>0)$ has vertices on the $x$–axis and axis…

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A family of parabolas $y=ax^{2}$ $(a>0)$ has vertices on the $x$–axis and axis parallel to $y$–axis. The differential equation of this family is:

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$y=ax^{2}$, $a=\dfrac{y}{x^{2}}$. Differentiate: $y'=2ax=\dfrac{2y}{x}$. $\Rightarrow x,y'-2y=0$.

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