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DE of All Parabolas with Axis Parallel to y-axis | JEE

JEE Maths question with a full step-by-step solution.

Question
The differential equation of all parabolas with axis parallel to the yy-axis is:
Ad2ydx2=2y+x\dfrac{d^2y}{dx^2} = 2y + x
Bd3ydx3=2y\dfrac{d^3y}{dx^3} = 2y
C(d2ydx2)3=dydx\left(\dfrac{d^2y}{dx^2}\right)^3 = \dfrac{dy}{dx}
Dd3ydx3=0\dfrac{d^3y}{dx^3} = 0correct
Solution
A parabola with axis parallel to the yy-axis has the form
x2+ax+by+c=0,x^2 + ax + by + c = 0,
 with three arbitrary constants a,b,ca, b, c, so we differentiate three times.
2x+a+bdydx=02x + a + b\frac{dy}{dx} = 0
2+bd2ydx2=02 + b\frac{d^2y}{dx^2} = 0
bd3ydx3=0⇒d3ydx3=0.b\frac{d^3y}{dx^3} = 0 \Rightarrow \frac{d^3y}{dx^3} = 0.
 Correct answer: (4)
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