How to solve the following diferential equation using variable separation method: y' = [ x (x² + 1) ] / 4y³.

With the initial value y(0) = -1 / sqrt(2), the solution is: y = -sqrt[ (x² + 1) / 2 ].
How to solve it?

y' = [x(x² + 1)]/(4y³)

dy/dx = (x)(x² + 1)/(4y³)

(4y³) dy = (x)(x² + 1) dx

Integrating both sides:

y^4 = x^4/4 + x²/2 + C

y =+/- (x^4/4 + x²/2 + C)^(1/4)

-1/√2 = +/- C^(1/4)

C = 1/4

y = - (x^4/4 + x²/2 + 1/4)^(1/4)

y = - ((x²/2 + 1/2)²)^(1/4)

y = -√[(x² + 1)/2]