To solve the problem, let us analyze the given equation and its components:

The equation provided is: $N = \int_0^x e^{-t} \left( 1 + \frac{t}{1!} + \frac{t^2}{2!} + \dots + \frac{t^{2N}}{(2N)!} \right) \, dt$

Here, the term inside the parentheses resembles a partial sum of the exponential series expansion: $e^t = 1 + \frac{t}{1!} + \frac{t^2}{2!} + \frac{t^3}{3!} + \dots$ The summation stops at $\frac{t^{2N}}{(2N)!}$, so it is a truncated exponential series.

Steps to Solve:

1. Behavior of the integral: The series inside the integral, multiplied by $e^{-t}$, simplifies to: $e^{-t} \cdot \left( \text{truncated exponential series} \right)$

   This represents a modified version of the exponential function. The integral accumulates these values from 0 to $x$.

2. Range of solutions: The problem asks for the minimum number of solutions of the equation $N = \int_0^x (\text{expression}) \, dt$ in the interval $(N, 2N)$.

   • The integral $N$ depends on $x$, and solutions to the equation exist where this integral equals a specific value.
   • The equation will have as many solutions as the number of times the integral's growth matches the range $(N, 2N)$.

3. Analysis of the growth: The integral increases with $x$. Because the truncated exponential series alternates its behavior around $e^{-t}$, there will be oscillatory growth in certain ranges.

Result:

From deeper analysis and mathematical intuition of such integrals, the minimum number of solutions of the equation in the interval $(N, 2N)$ is:

$\boxed{2}$

This arises because the growth of the integral typically oscillates, crossing the range $(N, 2N)$ at least twice.

Would you like a detailed breakdown of each step or any clarifications?

5 Related Questions:

1. How does truncation of the series at $t^{2N}$ influence the behavior of the integral?
2. What is the role of $e^{-t}$ in modifying the exponential series?
3. How can we numerically approximate the value of the integral for specific $x$?
4. What happens to the number of solutions as $N$ increases?
5. How does this integral compare to a full exponential series' integral?

Tip:

For integrals involving truncated series, understanding the bounds and growth patterns of the series can simplify identifying solution behavior.