The differential equation of all parabolas having vertex at the origin and axis along positive…
The differential equation of all parabolas having vertex at the origin and axis along positive $\mathrm{Y}$-axis is
$x^2 \frac{d y}{d x}-y=0$
$x \frac{d y}{d x}-2 y=0$
$x \frac{d y}{d x}+y=0$
$2 x \frac{d y}{d x}-y=0$
Solution
The equation of the parabola having vertex at origin and axis along the positive direction of \(y\)-axis is given by
\(x^2=4 a y \quad \ldots (1)\)
Since there is only one parameter, so we differentiate it only once.
Differentiating with respect to \(x\), we get
\(\begin{aligned}
& 2 x=4 a y^{\prime} \\
& \Rightarrow 4 a=\frac{2 x}{y^{\prime}}
\end{aligned}\)
Substituting the value of \(4 a\) in (1), we get
\(\begin{aligned}
& x^2=\frac{2 x}{y^{\prime}} \times y \\
& \Rightarrow x y^{\prime}=2 y \\
& \Rightarrow x y^{\prime}-2 y=0
\end{aligned}\)