The differential equation of all parabolas,wh | vedclass

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(C) The general equation of a parabola with its axis parallel to the $Y$-axis is given by $(x-h)^2 = 4a(y-k)$,where $(h, k)$ is the vertex and $a$ is

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$y_3=0$

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(C) The general equation of a parabola with its axis parallel to the $Y$-axis is given by $(x-h)^2 = 4a(y-k)$,where $(h, k)$ is the vertex and $a$ is a constant.This equation contains three arbitrary constants: $h$,$k$,and $a$.To find the differential equation,we differentiate with respect to $x$ three times.First differentiation: $2(x-h) = 4a \frac{dy}{dx} \implies (x-h) = 2a y_1$.Second differentiation: $1 = 2a y_2$.Third differentiation: $0 = 2a y_3$.Since $2a 
eq 0$,we must have $y_3 = 0$.

The differential equation of all parabolas,whose axes are parallel to the $Y$-axis,is

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