Question

The differential equation of all parabolas whose axes are parallel to y axis is

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

Answer 1

Answer: (A)

$\frac{d^{3}y}{dx^{3}} = 0$

Answer 2

The equation of a parabola whose axis is parallel to y-axis may be expressed as

$(x - \alpha)^{2} = 4a(y - \beta)$ ……..(i)

There are three arbitrary constants α, β and a.

We need to differentiate (i) 3 times

Differentiating (i) w.r.t. x, $2(x - \alpha) = 4a\frac{dy}{dx}$

Again differentiating w.r.t. x,

$2 = 4a\frac{d^{2}y}{dx^{2}}$ ⇒ $\frac{d^{2}y}{dx^{2}} = \frac{1}{2a}$

Differentiating w.r.t. x,

$\frac{d^{3}y}{dx^{3}} = 0$