The problem you provided involves solving a nonlinear initial value problem using a "reduced differential transform." This method applies to differential equations, particularly for nonlinear cases where traditional techniques can be cumbersome.

The given equation is:

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

with initial conditions:

$y(0) = 0, \quad y'(0) = 1.$

Steps to apply a Reduced Differential Transform Method (DTM):

1. Transforming the Equation: Differential Transform is based on a Taylor series expansion. It transforms the original differential equation into a simpler algebraic form in terms of power series coefficients. For a function $y(x)$, the differential transform $Y(k)$ of order $k$ is defined as:

   $Y(k) = \frac{1}{k!} \left[\frac{d^k y(x)}{dx^k}\right]_{x=0}$

   This converts the differential equation into a recurrence relation.

2. Applying the Transform: Let's start by applying the transform to each term of the equation $y''(x) + 2(y'(x))^2 + 8y(x) = 0$.

   • The second derivative $y''(x)$ transforms as $Y(k+2) = (k+2)(k+1)Y(k)$.

   • The nonlinear term $2(y'(x))^2$ becomes more complicated. Using convolution for product terms in differential transforms:

     $2(y'(x))^2 \rightarrow 2 \sum_{m=0}^{k} Y'(m)Y'(k-m)$

   • The linear term $8y(x)$ transforms as $8Y(k)$.

3. Initial Conditions: The initial conditions are: $Y(0) = y(0) = 0, \quad Y(1) = y'(0) = 1.$ These provide starting values for solving the recurrence relations.

4. Recurrence Relation: Using the transformed terms, you can derive a recurrence relation for the coefficients $Y(k)$ and solve iteratively for $Y(2), Y(3), Y(4), \dots$.

   After solving for a sufficient number of terms, the inverse differential transform is applied, yielding an approximate or exact solution in terms of a power series.

Would you like me to continue with the exact recurrence relation for this problem and calculate the series expansion of $y(x)$?

Here are some related questions:

1. Would you like a deeper explanation on how convolution works in DTM?
2. Do you want me to calculate specific terms of the series expansion $y(x)$?
3. Are you interested in how to handle nonlinear terms like $2(y'(x))^2$ in DTM?
4. Would you like to compare this approach to other numerical methods like Euler or Runge-Kutta?
5. Do you need clarification on the initial conditions and their role in solving the problem?

Tip: The reduced differential transform method is very efficient for solving nonlinear differential equations, especially when analytical methods struggle with complex terms like squared derivatives.