The differential equation of the family of parabolas with focus at the origin and the x–axis as axis is

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

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

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

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

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Solution

The correct option is B $y {(\frac{d y}{d x})}^{2} + 2 x \frac{d y}{d x} = y$

Equation of family of parabolas with focus at (0,0) and x-axis as axis is $y^{2} = 4 a (x + a)$ Differentiating (i) with respect to x,
$2 y y_{1} = 4 a, y^{2} = 2 y y_{1} (x + \frac{y y_{1}}{2}) y = 2 x y_{1} + y y_{1}^{2} \Rightarrow y {(\frac{d y}{d x})}^{2} + 2 x \frac{d y}{d x} = y$

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The differential equation of the family of parabolas with focus at the origin and the x–axis as axis is

The correct option is B y(dydx)2+2xdydx=y Equation of family of parabolas with focus at (0,0) and x-axis as axis is y2=4a(x+a) Differentiating (i) with respect to x, 2yy1=4a,y2=2yy1(x+yy12)y=2xy1+yy21⇒y(dydx)2+2xdydx=y