Find the particular solution, satisfying the given condition, for… - Vidyadip

Question

Find the particular solution, satisfying the given condition, for the following differential equation:$\frac{\text{dy}}{\text{dx}} - \frac{\text{y}}{\text{x}} + \text{cosec} \bigg(\frac{\text{y}}{\text{x}}\bigg) = \text {0; y = 0 when x} = 1.$

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Answer

$\text{Putting} \frac{\text{y}}{\text{x}} = \text{v}\Rightarrow \text{y = vx} \Rightarrow \frac{\text{dy}}{\text{dx}} = \text{v + x} \frac{\text{dv}}{\text{dx}}$$\therefore \text{we get v + x} \frac{\text{dv}}{\text{dx}} -\text{v} + \text{cosec} \text{v} = \text{o}$
$\Rightarrow \sin \text{v dv} + \frac{\text{dx}}{\text{x}} = \text{o}$
$\Rightarrow - \cos \text{v} + \log|\text{x}| = \text{c}_{1} \text{or} \cos \frac{\text{y}}{\text{x}} = \log|\text{x}| + \text{c}$
$\text{When x = 1, y = o} \Rightarrow \text{c} = 1$
$\text{Hence the solution is} \cos \frac{y}{x} = 1 + \log|x|$

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