If y= ⁻¹ ( 1+a^2 x^2 -1 a x ) , then (1+a^2 x^2 ) y^ ^ +2 a^2 x y^ is equal to

Step-by-step solution

Given, y= ⁻¹ ( 1+a^2 x^2 -1 a x ) Put a x= aligned y & = ⁻¹ ( 1+ ^2 -1 ) & = ⁻¹ ( -1 ) & = ⁻¹ ( 1- ) & = ⁻¹ ( 2 ^2 2 2 2 2 ) & = ⁻¹ ( ^ 2 ) & = 2 = 1 2 ⁻¹ a x y & = 1 2 ⁻¹ a x aligned On differentiating w.r.t. x , we get y^ = 1 2 (1+a^2 x^2 ) Again, differentiating both side we get aligned y^ & =- 1 2 (a^2 2 x ) (1+a^2 x^2 )^2 (1+a^2 x^2 ) y^ & =- a^2 x 1+a^2 x^2 & =-a^2 x (2 y^ ) [ from Eq. (i) ] (1+a^2 x^2 ) y^ & +2 a^2 x y^ =0 aligned