JEE Main · 2021 · 18 Mar 2021 · Shift 1 · Differential Equations

The differential equation satisfied by the system of parabolas

y^{2} = 4 a (x + a)

is :

Answer: (A)

Solution

Differentiating

y^{2} = 4 a (x + a)

gives

2 y \frac{d y}{d x} = 4 a

, so

a = \frac{y}{2} \frac{d y}{d x}

. Substituting this back,

y^{2} = 4 \cdot \frac{y}{2} \frac{d y}{d x} (x + \frac{y}{2} \frac{d y}{d x}) = 2 x y \frac{d y}{d x} + y^{2} (\frac{d y}{d x})^{2}

Dividing by

y

and collecting terms gives

y (\frac{d y}{d x})^{2} + 2 x \frac{d y}{d x} - y = 0

JEE Main · 2021 · 18 Mar 2021 · Shift 1 · Single correct option

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