The given differential equation is:

$\frac{dy}{dx} + \frac{1}{x}y = y^5$

with the initial condition $y(0) = 1$.

Solution:

This is a first-order nonlinear differential equation. However, the initial condition $y(0) = 1$ appears problematic because the term $\frac{1}{x}$ is undefined at $x = 0$. As such, the initial condition should be reconsidered unless there is a typo.

If you intended a different initial condition, such as $y(1) = 1$, let me know so we can proceed. Otherwise, I will solve the equation generally.

General Approach:

This equation is not linear or separable in its current form, so we cannot solve it directly without further techniques like numerical methods or approximations.

Let me know how you'd like to proceed or clarify the initial condition so we can delve further into the solution.

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Questions for deeper understanding:

1. Should the initial condition be reconsidered?
2. Are numerical methods acceptable for solving this?
3. Would you like to explore stability and existence for this problem?
4. Should we attempt a power series or approximation method?
5. Is there a specific method you'd like me to use (e.g., Laplace)?

Tip: Always double-check initial conditions in differential equations for domain constraints!