1. What is Euler's Method?

Euler's method is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explicit method for numerical integration of ordinary differential equations.

2. How Does the Calculator Work?

The calculator uses Euler's method formula:

Where:

• \( y_n \) — Current y-value
• \( h \) — Step size
• \( f(x_n, y_n) \) — Function value at current point
• \( y_{n+1} \) — Next y-value approximation

Explanation: The method approximates the solution by taking small steps along the tangent line at each point.

3. Importance of Euler's Method

Details: Euler's method is fundamental in numerical analysis and provides a simple way to approximate solutions to differential equations when analytical solutions are difficult or impossible to obtain.

4. Using the Calculator

Tips: Enter the current y-value (y_n), step size (h), and the function value f(x_n, y_n). The step size must be positive for accurate results.

5. Frequently Asked Questions (FAQ)

Q1: What are the limitations of Euler's method?
A: Euler's method has relatively low accuracy and can accumulate significant errors over many steps, especially for stiff equations.

Q2: When should I use Euler's method?
A: It's suitable for simple problems and educational purposes, but for more accurate results, higher-order methods like Runge-Kutta are preferred.

Q3: How does step size affect accuracy?
A: Smaller step sizes generally improve accuracy but require more computation. Very small steps may lead to rounding errors.

Q4: Can Euler's method handle systems of equations?
A: Yes, Euler's method can be extended to systems of first-order differential equations.

Q5: What is the local truncation error?
A: The local truncation error for Euler's method is proportional to h², meaning it's a first-order method.