The solution of the differential equation y d y d x =x [ y² x² + ( y² x² ) ^ ( y² x² ) ] is (where, C is a constant)

Step-by-step solution

Let v = y^2 x^2 . Then y^2 = v x^2 . Differentiating with respect to x , we get 2y dy dx = v(2x) + x^2 dv dx , which implies y dy dx = vx + x^2 2 dv dx . Substituting this into the given differential equation y dy dx = x [ v + (v) '(v) ] , we have vx + x^2 2 dv dx = vx + x (v) '(v) . Simplifying the equation, we obtain x^2 2 dv dx = x (v) '(v) , which simplifies to x 2 dv dx = (v) '(v) . Rearranging the terms to separate variables, we get '(v) (v) dv = 2 x dx . Integrating both sides, we have '(v) (v) dv = 2 x dx ,