Let y=y (x ) be the solution of the differentialequation xdy- ydx= (x^2-y^2) dx ,x 1, with y (1 )=0. If the area bounded by the line x=1 ,x=e^ ,y=0 and y=y (x ) is e^2 + , then the value of 10 ( + )…

Question

Let y=y (x ) be the solution of the differentialequation xdy- ydx= (x^2-y^2) dx ,x 1, with y (1 )=0. If the area bounded by the line x=1 ,x=e^ ,y=0 and y=y (x ) is e^2 + , then the value of 10 ( + ) is equal to_____

Accepted Answer

Correct answer: 4. (4) xdy- ydx= x^2-y^2 dx ⇒ xdy- ydxx^2= 1x 1- y^2x^2 dx ⇒ d ( yx ) 1- ( yx )^2= dxx ⇒ sin^-1 ⁡ ( yx )=ln |x |+c at x=1 ,y=0 ⇒c=0 y=x sin ⁡ (l nx ) A= 0.05emx sin ⁡ (l nx ) dx a=e^t , dx=e^t dt ⇒ 0.05eme^2t sin ⁡ (t ) dt=A ⇒ a= 15 , = 15 so10 ( + )=4

$Let y = y (x) be the solution of the differentialequation xdy - ydx = (x^{2} - y^{2}) dx, x \geq 1, with y (1) = 0 . If the area bounded by the line x = 1, x = e^{π}, y = 0 and y = y (x) is α e^{2 π} + β, then the value of 10 (α + β) is equal to_____$

Solution

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