Find differential equation of the family of curves $y^{2} = 4 a x$

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Solution

Consider the given equation.

$y^{2} = 4 a x$ …… (1)

On differentiating both sides w.r.t $x$ , we get

$2 y \frac{d y}{d x} = 4 a$ ……. (2)

From equations (1) and (2), we get

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

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

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

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

Hence, this is the answer.

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The differential equation of family of curves $y^{2} = 4 a (x + a)$ is
a) $y^{2} = 4 \frac{d y}{d x} (x + \frac{d y}{d x})$
b) $2 y \frac{d y}{d x} = 4 a$
c) $y \frac{d^{2} y}{d x^{2}} + {(\frac{d y}{d x})}^{2} = 0$
d) $2 x \frac{d y}{d x} + y {(\frac{d y}{d x})}^{2} - y = 0$

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Find differential equation of the family of curves y2=4ax

Consider the given equation. y2=4ax …… (1) On differentiating both sides w.r.t x, we get 2ydydx=4a ……. (2) From equations (1) and (2), we get y2=2ydydxx 2xdydx=y dydx=y2x dydx−y2x=0 Hence, this is the answer.

Find differential equation of the family of curves $y^{2} = 4 a x$