Find the particular solution, satisfying the given condition, for the following differential equation $\frac{d y}{d x} - \frac{y}{x} + c o s e c (\frac{y}{x}) = 0; y = 0$ when $x = 1$ .

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Solution

Now,
$\frac{d y}{d x} - \frac{y}{x} + c o s e c (\frac{y}{x}) = 0$
$y = v x$
$\frac{d y}{d x} = v + x \frac{d v}{d x}$
$v + x \frac{d v}{d x} - v + cos e c v = 0$
$x \frac{d v}{d x} = - cos e c v$
$\int - sin v d v = \int \frac{1}{x} d x + C$
$cos v = log x + C$
$cos (\frac{y}{x}) = log x + C$
$y = 0 w h e n x = 1$
$1 = c$
$∴ cos (\frac{y}{x}) = log x + 1$

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