Let y = y ( x ) be the solution of the differential equation ( x 2 – 3 y 2 ) dx + 3 xy dy = 0 , y ( 1 ) = 1 . Then 6 y 2 e is equal to

Step-by-step solution

Given equation is: ( x 2 –   3 y 2 ) dx + 3 xydy = 0 We can re-write equation as dy dx = - x 2 - 3 y 2 3 xy ⇒ dy dx = y x - 1 3 x y   ....(1) Put y = vx dy dx = v + x dv dx Equation (1) can be written as v + x dv dx = v - 1 3 1 v ⇒ vdv = - 1 3 x Integrating both sides, we get v 2 2 = - 1 3 ln x + c ⇒ y 2 2 x 2 = - 1 3 ln x + c .....(2) ∵   y 1 = 1 (given) ∴   c = 1 2 (from equation 2) Equation (2) can be written as y 2 2 x 2 = - 1 3 ln x + 1 2 ⇒ y 2 = -