The differential equation of all parabolas having their axis of symmetry with the axis of x is?

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

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

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

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

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Solution

The correct option is D $y = 2 x \frac{d y}{d x}$
$L e t a i s c o n s t a n t y^{2} = 4 a x \to (i) d i f f e r e n t i a t e w i t h r e s p e c t i o n \Rightarrow 2 y \frac{d y}{d x} = 4 a \times 1 \Rightarrow 2 y \frac{d y}{d x} = 4 a \to (i i) P u t 4 a = 2 y \frac{d y}{d x} i n e q^{n} (i) \Rightarrow y^{2} = 2 y \frac{d y}{d x} \times x \Rightarrow y = 2 \frac{d y}{d x} \times x ∴ y = 2 x \frac{d y}{d x} I t i s r e q u i r e d d i f f e r e n t i a t e e q^{n} o f p a r a b o l a$

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The differential equation of all parabolas having their axis of symmetry with the axis of x is?

The correct option is D y=2xdydxLetaisconstanty2=4ax→(i)differentiatewithrespection⇒2ydydx=4a×1⇒2ydydx=4a→(ii)Put4a=2ydydxineqn(i)⇒y2=2ydydx×x⇒y=2dydx×x∴y=2xdydxItisrequireddifferentiateeqnofparabola