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Question
Solve the following initial value problem:
\[x\frac{dy}{dx} - y + x \sin\left( \frac{y}{x} \right) = 0, y\left( 2 \right) = x\]
Solution
\[x\frac{dy}{dx} - y + x \sin\left( \frac{y}{x} \right) = 0, y(2) = \pi\]
It is a homogeneous equation . put y = vx
\[\text{ and }\frac{dy}{dx} = v + x\frac{dv}{dx}\]
\[\text{ so,} v + x\frac{dv}{dx} = \frac{vx}{x} - \sin\left( \frac{vx}{x} \right)\]
\[x\frac{dv}{dx} = - \sin v\]
\[\frac{dv}{\sin v} = - \frac{dx}{x}\]
\[\text{ cosec }(v)dv = - \frac{dx}{x}\]
Integraing both sides we get,
\[\log(\text{cosec }(v) - cot(v)) = - \log x + \log c\]
\[log\left( \text{cosec }\left( \frac{y}{x} \right) - cot\left( \frac{y}{x} \right) \right) = - log x + log c\]
\[\text{Putting the values }x = 2\text{ and }y = \pi \]
\[log\left(\text{cosec }\left( \frac{\pi}{2} \right) - cot\left( \frac{\pi}{2} \right) \right) = - log 2 + log c\]
\[c = 0\]
\[log\left( \text{cosec }\left( \frac{y}{x} \right) - cot\left( \frac{y}{x} \right) \right) = - log x\]
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