Question

The differential equation of all parabolas having vertex at the origin and axis along positive $Y$-axis is

• A. $x^2 \frac{d y}{d x}-y=0$
• B. $x \frac{d y}{d x}+2 y=0$
• C. $x \frac{d y}{d x}=2 y$
• D. $x^3 \frac{d y}{d x}=3 y$

Answer 1

Answer: (C)

The general equation of an parabola having vertex at the origin and axis along positive $Y$-axis is
$x^2=4 a y \ldots \text { (i) }$
On differentiating Eq. (i), we get
$ 2 x=4 a \frac{d y}{d x}$
$\Rightarrow \quad \frac{d y}{d x}=\frac{x}{2 a} \Rightarrow 2 a=\frac{x}{d y / d x}$
Putting value of $2 a$ in Eq. (i), we get
$x^2=2\left(\frac{x}{d y / d x}\right) y \Rightarrow x \frac{d y}{d x}=2 y$