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Question
Show that the given differential equation is homogeneous and solve them.
`x dy/dx - y + x sin (y/x) = 0`
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Solution
`x dy/dx - y + x sin (y/x) = 0`
`dy/dx = y/x - sin y/x = g (y/x)` (say) ....(i)
The right side of the equation is in the form of `g(y/x)` so it is a homogeneous differential equation of zero degree.
∴Putting y = vx
`dy/dx = v + x dy/dx` From equation (i)
`=> v + x (dv)/dx = v - sin v`
`=> x (dv)/dx = v - sin v - v`
`=> x (dv)/dx = - sin v`
`=> cosec v dv = - 1/x dx`
⇒ log |cosec v - cot v|
= - log |x| + C1
On integrating,
⇒ log |(cosec v - cot v)| = C1
⇒ |x (cosec v - cot v)| = eC1
⇒ x (cosec v - cot v) = ± eC1 = C (say)
⇒ `x (cosec y/x - cot y/x) = C`
Which is the required general solution.
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