Euler's Method is a numerical technique used to approximate solutions to ordinary differential equations (ODEs). It works by taking small steps from an initial condition and using the slope of the differential equation to estimate the value of the function at the next step. The formula for Euler's Method is:

$y_{n+1} = y_n + h \cdot f(x_n, y_n)$

where:

• $y_n$ is the current value of the function,
• $f(x_n, y_n)$ is the derivative (or slope) at $(x_n, y_n)$,
• $h$ is the step size (the difference between consecutive $x$-values).

Smaller Step Size (h) and Accuracy

A smaller value of $h$ results in a better approximation because it reduces the error at each step. Since Euler's Method is based on a linear approximation of the curve, larger step sizes cause more deviation from the actual solution. By reducing $h$, the approximation follows the curve more closely, minimizing the local truncation error.

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Here are 5 related questions to expand on this:

1. What is the global error in Euler's method, and how is it calculated?
2. How does Euler's method compare to other numerical methods like the Runge-Kutta methods?
3. Can Euler's method handle systems of differential equations?
4. What are the limitations of Euler's method in terms of stability?
5. How can we modify Euler’s method to improve accuracy further?

Tip: If accuracy is critical, smaller step sizes improve results but also increase computational time. Consider using adaptive step size methods for balance.