The differential equation of all parabolas wh | vedclass

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Question

(A) The general equation of a parabola with its axis along the $y$-axis is given by $(x - 0)^2 = 4a(y - k)$,where $a$ and $k$ are arbitrary constants.

Vedclass · Accepted Answer

$x \frac{d^2y}{dx^2} - \frac{dy}{dx} = 0$

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(A) The general equation of a parabola with its axis along the $y$-axis is given by $(x - 0)^2 = 4a(y - k)$,where $a$ and $k$ are arbitrary constants.This simplifies to $x^2 = 4ay - 4ak$.Differentiating both sides with respect to $x$,we get:$2x = 4a \frac{dy}{dx}$$\Rightarrow x = 2a \frac{dy}{dx}$$\Rightarrow \frac{1}{2a} = \frac{1}{x} \frac{dy}{dx}$Differentiating again with respect to $x$ to eliminate the arbitrary constant $a$:$\frac{d}{dx} \left( \frac{1}{x} \cdot \frac{dy}{dx} \right) = \frac{d}{dx} \left( \frac{1}{2a} \right)$Using the product rule on the left side:$\frac{1}{x} \cdot \frac{d^2y}{dx^2} + \frac{dy}{dx} \left( - \frac{1}{x^2} \right) = 0$Multiplying the entire equation by $x^2$:$x \frac{d^2y}{dx^2} - \frac{dy}{dx} = 0$

The differential equation of all parabolas whose axis is the $y$-axis is

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