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Question
Find the general solution of the following differential equation:
`x (dy)/(dx) = y - xsin(y/x)`
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Solution
We have the differential equation:
`(dy)/(dx) = y/x - sin(y/x)`
The equation is a homogeneous differential equation.
Putting `y = vx ⇒ (dy)/(dx) = v + x (dv)/(dx)`
The differential equation becomes
`v + x (dv)/(dx) = v - sinv`
⇒ `(dv)/(sinv) = - (dx)/x` ⇒ cosecvdv = `-(dx)/x`
Integrating both sides, we get
`log|"cosec"v - cotv| = - log|x| + logK, K > 0` (Here, log K is an arbitrary constant.)
⇒ `log|("cosec"v - cotv)x| = log K`
⇒ `|("cosec"v - cotv)x| = K`
⇒ `("cosec"v - cotv)x = +- K`
⇒ `("cosec" y/x - cot y/x)x = C`, which is the required general solution.
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