Question

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

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

Answer 1

Answer: (C)

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