1. **State the problem:** We need to solve the differential equation $$G'[\Xi] = \Delta (\Lambda G[\Xi]^3 - G[\Xi]) + \mu.$$\n\n2. **Rewrite the equation:** The equation can be written as $$\frac{dG}{d\Xi} = \Delta \Lambda G^3 - \Delta G + \mu.$$\n\n3. **Separate variables if possible:** We consider $\Xi$ as the independent variable and $G$ as the dependent variable and rewrite: $$\frac{dG}{d\Xi} = \Delta \Lambda G^3 - \Delta G + \mu.$$\n\n4. **Solve the separable ODE:** Rewrite as $$\frac{dG}{\Delta \Lambda G^3 - \Delta G + \mu} = d\Xi.$$\n\n5. **Integrate both sides:** We need to integrate $$\int \frac{dG}{\Delta \Lambda G^3 - \Delta G + \mu} = \int d\Xi = \Xi + C.$$\n\n6. **Integration complexity:** The integral on the left is non-trivial because the denominator is a cubic polynomial in $G$. Solving this integral involves partial fraction decomposition after factorization or using advanced techniques depending on the discriminant of the cubic polynomial.\n\n7. **General implicit solution:** The solution is implicit in terms of the integral: $$\Xi + C = \int \frac{dG}{\Delta \Lambda G^3 - \Delta G + \mu}.$$\n\n8. **Summary:** The explicit solution $G[\Xi]$ generally requires solving the integral above, which depends on parameters $\Delta$, $\Lambda$, and $\mu$ and may not have a closed-form solution for all values. Numerical methods or special functions might be used for specific values.