Let y = f(x) be a curve passing through `(e,e^( e ))` which satisfy the differential equation
` ( 2ny + xy log_(e) x ) dx - x log_(e) x dy = 0 , x gt 0, y gt 0`

Options

A. e
B. 1
C. 0
D. 2

Answer

c

Explanation

`2ny + xy log _(e)x) dx = x log_(e) x dy`
` rArr (dy)/y = ((2n)/(x log _(e)x)+1) dx`
` rArr log (y) = 2n log | log x| + x+C`
` :. " Curve through" (e,e^(e )) ` so , C = 0
` :. y = e^(x+log(logx)2n)`
` rArr f(x) = e^(x) (log x )^(2n) `
Now, `g(x) =underset(nrarroo)lim f(x)={{:(oo","," if",xlt(1)/(e)),(0","," if ",(1)/(e)lt xlt e),(oo","," if ",x gt e):}`
`therefore" "int_(1//e)^(e)g(x)dx=0`