respect to \( x \), we get

  \[
    -y^{-1} = \frac{x^3}{3} + c
  \]

  where \( c \) is the constant of integration.  Taking the negative
  reciprocal of both sides, we get

  \[
    y = \frac{-3}{x^3 + 3c} = \frac{-3}{x^3 + d}.
  \]

  where \( d = 3c \).
</p>
<p>
  This is indeed a valid solution of the differential equation.
  However, we have lost a solution.  The lost solution is
  \( y = 0 \).  We can verify that with \( y = 0 \), we get \( 0 \) on
  both sides of the differential equation.  We lost