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NDA 2023 Maths · Mathematics · Differential Equations · hard

The differential equation of the family of parabolas y^2=4ax, where a is an arbitrary constant, is

Single correct option

The differential equation of the family of parabolas y2=4axy^2=4ax, where aa is an arbitrary constant, is
  1. A. y=2xdydxy=2x\dfrac{dy}{dx}
  2. B. y=xdydxy=x\dfrac{dy}{dx}
  3. C. ydydx=2xy\dfrac{dy}{dx}=2x
  4. D. dydx=2yx\dfrac{dy}{dx}=\dfrac{2y}{x}
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Answer: (A) y=2xdydxy=2x\dfrac{dy}{dx}

Differentiating gives 2ydydx=4a2y\dfrac{dy}{dx}=4a. Substituting 4a=y2x4a=\dfrac{y^2}{x} yields 2ydydx=y2x2y\dfrac{dy}{dx}=\dfrac{y^2}{x}, i.e. y=2xdydxy=2x\dfrac{dy}{dx} after cancelling yy. The option dydx=2yx\dfrac{dy}{dx}=\dfrac{2y}{x} is the equation of y=cx2y=cx^2.
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Original question written in the NDA 2023 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake