The equation you've provided:

∑i=1nAiEit=b\sum_{i=1}^{n} A_i E_i^t = b

is a non-linear transcendental equation due to the presence of the exponential terms EitE_i^t. These types of equations don't generally have closed-form solutions (except in special cases), and typically, numerical methods are the way to go for finding roots.

1. Classification of the Equation:

This equation doesn't fall into a specific class with a simple closed-form solution but can be classified as a sum of exponentials equation. While these are common in various fields like physics and engineering (e.g., in decay processes, financial models), solving them analytically is often not feasible.

2. Analytical Methods:

For this equation, analytical methods are generally not available unless nn is very small, and even then, it would be rare to find a closed-form expression. If n=1n = 1, the equation becomes:

A1E1t=bA_1 E_1^t = b

which can be solved for tt explicitly by taking the natural logarithm:

t=ln⁡(b/A1)ln⁡(E1)t = \frac{\ln(b/A_1)}{\ln(E_1)}

However, for general n>1n > 1, there is no simple closed-form solution due to the sum involving multiple exponential terms.

3. Numerical Methods:

Since analytical solutions are unlikely, numerical methods are the most practical way to solve for tt. The challenge you face, as you mentioned, is to find the smallest positive root. Let’s discuss a few numerical methods that could work:

3.1. Bisection Method:

This method works by repeatedly narrowing down the interval in which the root exists. It requires two initial points, t1t_1 and t2t_2, where the function changes sign, i.e., f(t1)⋅f(t2)