The differential equation of the family of pa | vedclass

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(B) The standard equation of a parabola with focus at $(0,0)$ and axis along the $X$-axis is given by $(x+a)^2 + y^2 = (x+2a)^2$,which simplifies to $

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$-y\left(\frac{dy}{dx}\right)^2=2x\frac{dy}{dx}-y$

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(B) The standard equation of a parabola with focus at $(0,0)$ and axis along the $X$-axis is given by $(x+a)^2 + y^2 = (x+2a)^2$,which simplifies to $y^2 = 4a(x+a)$. However,for a parabola with focus at the origin and axis as the $X$-axis,the equation is $y^2 = 4a(x+a)$.Let the parabola be $y^2 = 2a(x + a/2)$,or more simply $y^2 = 2ax + a^2$ where $a$ is the parameter.Differentiating with respect to $x$:$2y \frac{dy}{dx} = 2a \Rightarrow a = y \frac{dy}{dx}$.Substituting $a$ into the original equation $y^2 = 2ax + a^2$:$y^2 = 2x(y \frac{dy}{dx}) + (y \frac{dy}{dx})^2$.Dividing by $y$ (assuming $y 
eq 0$):$y = 2x \frac{dy}{dx} + y \left(\frac{dy}{dx}ight)^2$.Rearranging the terms:$y \left(\frac{dy}{dx}ight)^2 = y - 2x \frac{dy}{dx}$,which is equivalent to $-y \left(\frac{dy}{dx}ight)^2 = 2x \frac{dy}{dx} - y$.

The differential equation of the family of parabolas with focus at the origin and the axis as the $X$-axis is:

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