Question

Find the general solution of the differential equation `x (dy)/(dx) = y(logy - logx + 1)`.

Answer 1

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