If f(x) x x d x= 1 2 (b²-a² ) f(x)+c, where c is the constant of integration, then f(x) is

Step-by-step solution

If f(x) x x d x= 1 2 (b²-a² ) f(x)+c aligned & LHS 1 2 f ( x ) (2 x x ) dx & = 1 2 f ( x ) 2 x dx & [ Here , put f ( x )= 2 ( ~b ²- a ² ) 1 2 x ] &= 1 2 2 ( ~b ²- a ² ) 2 x 2 x dx aligned aligned &= 1 (b²-a² ) 2 x d x &= 1 (b²-a² ) 2 x 2 +c₁ &= 1 2 (b²-a² ) ( 1 2 x )+c₁ aligned aligned & [ Put ₁= c + 1 2 ( ~b ²- a ² ) 2 ( ~b ²- a ² ) ] =& 1 2 ( ~b ²- a ² ) ( 1 2 x ) &+ 1 2 ( ~b ²- a ² ) ( 2 ( ~b ²- a ² ) )+ c =& 1 2 ( ~b ²- a ² ) [ 2 ( ~b ²- a ² ) 2 x ]+ c aligned Hence, f(x)= 2 (b²-a² ) 2 x