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The given problem involves solving a nonlinear second-order differential equation:

\[
y''(x) + 2(y'(x))^2 + 8y(x) = 0, \quad 0 \leq x < \infty
\]
with initial conditions:
\[
y(0) = 0 \quad \text{and} \quad y'(0) = 1.
\]

### Step-by-Step Approach:

1. **Reduce the order of the differential equation**: 

   Let \( z(x) = y'(x) \). Thus, the second derivative of \( y \), \( y''(x) \), can be written as:
   \[
   y''(x) = z'(x).
   \]

   Substituting into the original equation:
   \[
   z'(x) + 2z(x)^2 + 8y(x) = 0.
   \]

2. **Express \( y(x) \) in terms of \( z(x) \)**:

   We know that \( y'(x) = z(x) \), so \( y(x) = \int z(x) \, dx \).
   The equation now becomes:
   \[
   z'(x) + 2z(x)^2 + 8 \left( \int z(x) \, dx \right) = 0.
   \]

   This is a first-order nonlinear differential equation in \( z(x) \).

3. **Apply initial conditions**: 
   - From \( y(0) = 0 \), this implies:
     \[
     y(0) = \int z(x) \, dx \bigg|_{x=0} = 0.
     \]
   - From \( y'(0) = 1 \), we know that:
     \[
     z(0) = 1.
     \]

### Next Steps:

We now need to solve this reduced differential equation using either analytical methods (such as separation of variables or a known method for solving first-order nonlinear ODEs) or numerical techniques, depending on the complexity. 

Would you like a detailed exploration of the solution method, or would you prefer a numerical solution to this equation?

Consider the nonlinear initial value problem y''(x) + 2(y'(x))^2 + 8y(x) = 0, 0 <= x < infinity, with initial conditions y(0) = 0 and y'(0) = 1.

Explore how to solve a nonlinear initial value problem: y''(x) + 2(y'(x))^2 + 8y(x) = 0, with initial conditions y(0) = 0 and y'(0) = 1. This example demonstrates key methods for solving ordinary differential equations (ODEs), suitable for advanced undergraduate students.

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