Let y = f ( x ) be a curve passing through ( e , e e ) which satisfy the differential equation ( 2 n y + x y lo g e x ) d x − x lo g e x d y = 0 x 0 , y 0 . If g ( x ) = lim n → ∞ ( 0.020 )

Question

Let y = f ( x ) be a curve passing through ( e , e e ) which satisfy the differential equation ( 2 n y + x y lo g e ​ x ) d x − x lo g e ​ x d y = 0 x > 0 , y > 0 . If g ( x ) = lim n → ∞ ​ ( 0.020 ) f ( x ) . Then ∫ e 1 ​ e ​ g ( x ) dx equal to

TestHub · Accepted Answer

( 2 n y + x y lo g e ​ x ) d x = x lo g e ​ x d y ⇒ y d y ​ = ( x l o g e ​ x 2 n ​ + 1 ) d x ⇒ lo g ( y ) = 2 n lo g ∣ lo g x ∣ + x + c and c = 0 . Now, g ( x ) = lim n → ∞ ​ f ( x ) = ⎩ ⎨ ⎧ ​ → ∞ 0 → ∞ ​ i f x < e 1 ​ i f e 1 ​ < x < e i f x > e ​ ∫ 0 e ​ g ( x ) d x = 0 .

Mathematics - Differential Equation Question with Solution | TestHub

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