The given problem is a nonlinear second-order differential equation with initial conditions. The equation is:

$y''(x) + 2(y'(x))^2 + 8y(x) = 0, \quad 0 \leq x < \infty$

With the initial conditions:

$y(0) = 0 \quad \text{and} \quad y'(0) = 1.$

Reduced Differential Transform Method (DTM)

To solve this equation using the Differential Transform Method (DTM), we can follow these steps:

1. Apply the Differential Transform: The differential transform of the $n$-th derivative of $y(x)$ is given by: $Y(n) = \frac{1}{n!} \left. \frac{d^n y(x)}{dx^n} \right|_{x=0}$ where $Y(n)$ is the differential transform of $y(x)$.

2. Transform the given differential equation: Now, apply the differential transform to each term in the equation. The transform of $y''(x)$, $2(y'(x))^2$, and $8y(x)$ must be computed.

3. Substitute the initial conditions: Use the initial conditions $y(0) = 0$ and $y'(0) = 1$ to determine the coefficients $Y(0)$ and $Y(1)$.

4. Reconstruct the series solution: After computing the transformed terms and solving for the coefficients, the inverse differential transform is applied to obtain the series solution for $y(x)$.

Would you like me to continue with the detailed solution process using DTM?

Here are 5 relative questions to expand on this topic:

1. What is the general form of the Differential Transform Method (DTM) for higher-order differential equations?
2. How does DTM differ from other series methods, such as the Taylor series or power series method?
3. Can DTM handle boundary conditions as well, or only initial conditions?
4. What are the advantages of using DTM for nonlinear differential equations?
5. How does the convergence of DTM solutions compare to numerical methods?

Tip: When using DTM, ensure that the nonlinearity in the equation is handled carefully, often requiring transformations of product terms like $(y'(x))^2$.