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Question
Find the general solution of the differential equation `x (dy)/(dx) = y(logy - logx + 1)`.
Solution
Given differential equation is `x (dy)/(dx) = y(logy - logx + 1)`
⇒ `(dy)/(dx) = y/x(log y/x + 1)`
Put y = vx
⇒ `(dy)/(dx) = v + x(dv)/(dx)`
⇒ `v + x (dv)/(dx) = v(logv + 1)`
⇒ `(dv)/(vlogv) = (dx)/x`
On integrating both sides, we get
`int (dx)/x = int(dx)/x`
⇒ log(logv) = logx + logC
⇒ log(logv) = logCx
⇒ log(y/x) = Cx
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