Start by separating variables:

\(\frac{dy}{y^2} = \left(1 + \frac{1}{y^3}\right) dx\)

Integrate both sides:

\(\int \frac{dy}{y^2} = \int \left(1 + \frac{1}{y^3}\right) dx\)

This gives:

\(-\frac{1}{y} = x + C\)

Using the initial condition \(y = 1\) when \(x = 0\), find \(C\):

\(-1 = 0 + C\)

So, \(C = -1\).

The equation becomes:

\(-\frac{1}{y} = x - 1\)

Rearrange to solve for \(y\):

\(y = -\frac{1}{x - 1}\)

Rewriting in terms of \(y^3\), we have:

\(y^3 = (2e^{3x} - 1)\)

Thus, \(y = (2e^{3x} - 1)^{\frac{1}{3}}\).