Question

If `x (dy)/(dx) = y(log y - log x + 1)`, then the solution of the dx equation is

पर्याय

• `y log (x/y) = cx`

• `x log (y/x) = cy`

• `log (y/x) = cx`

• `log (x/y) = cy`

Answer 1

`log (y/x) = cx`

Explanation:

`x (dy)/(dx) = y(log y - log x + 1)`

⇒ `(dy)/(dx) = y/x (log y/x + 1)`

Put `y = vx`

⇒ `(dy)/(dx) = v + x (dc)/(dx)`

⇒ `v + x (dv)/(dx) = v(log v + 1)`

⇒ `(dv)/(v log v) = (dx)/x`

Integrating both sides,

⇒ `log(log v) = log x + log c = log cx`

⇒ `log v = cx`

⇒ `log  (y/x) = cx`