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Exponential Representation for Irrational Numbers

Find the valu...

Question

Find the value of $d y / d x$ ,
$y^{2} = 4 a x$

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Solution

Given,

$y^{2} = 4 a x$

differentiating on both sides, we get,

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

$2 y \frac{d y}{d x} = 4 a$

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

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

y^{2} = 4 a (x + a)

is the solution of differential equation

y \times [1 - {(\frac{d y}{d x})}^{2}] = 2 x \frac{d y}{d x}

y^{2} = 4 a (x + a)

is a solution of the differential equation

y {1 - {(\frac{d y}{d x})}^{2}} = 2 x \frac{d y}{d x}

y^{2} = 4 a x : y = x \frac{d y}{d x} + a \frac{d x}{d y}

\frac{d y}{d x}

:

y^{2} = a (b^{2} - x^{2})

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