Let y=f(x) be the solution of the differential equation d y ~d x + x y x^2-1 = x^6+4 x 1-x^2 ,-1 x 1 such that f(0)=0 . If 6 _ -1 / 2 ^ 1 / 2 f(x) d x=2 - then ^2 is equal to _______ .

Step-by-step solution

I.F. e ^ - 1 2 2 x 1- x ^2 dx = e ^ - 1 2 (1- x ^2 ) = 1- x ^2 y 1-x^2 = (x^6+4 x ) d x= x^7 7 +2 x^2+c Given y(0)=0 c=0y= x^7 7 +2 x^2 1-x^2 Now, 6 _ - 1 2 ^ 1 2 x^7 7 +2 x ^2 1- x ^2 dx =6 _ - 1 2 ^ 1 2 2 x ^2 1- x ^2 dx aligned & =24 ₀^ 1 2 x^2 1-x^2 d x & Put x= & d x= d & =24 ₀^ 6 ^2 d & =24 ₀^ 6 ( 1- 2 2 ) d =12 [ - 2 2 ]₀^ 6 & =12 ( 6 - 3 4 ) & =2 -3 3 & ^2=(3 3 )^2=27 aligned