Euler's Method Calculator
Use Euler's Method to estimate y(x) for dy/dx = f(x, y).
Table of Contents
How to Use
- Write the derivative f(x, y) using x, y, and functions like sin, cos, exp, ln, sqrt, or abs.
- Enter the initial point (x₀, y₀) that satisfies the differential equation.
- Choose a positive step size and how many iterations to take.
- Click Calculate to generate the Euler table and read the final approximation y(x₀ + n·h).
How Euler's Method Works
Euler's Method replaces the continuous differential equation with a sequence of tangent-line steps: yₙ₊₁ = yₙ + h · f(xₙ, yₙ). Smaller step sizes generally produce more accurate approximations but require more iterations.
Because the approach uses only the current slope, it can accumulate error quickly for rapidly changing derivatives. Compare different step sizes to judge stability.
Best Practices
- Keep h small relative to the curvature of the solution curve.
- Check results by halving the step size and comparing values.
- Ensure the derivative expression is continuous over the interval of interest.
- Use Euler's Method for initial intuition, then switch to Heun or Runge–Kutta for higher accuracy.
Frequently Asked Questions
- Can I enter trigonometric or exponential functions?
- Yes. The parser understands sin, cos, tan, exp, ln/log, sqrt, abs, and their hyperbolic versions. Always type expressions using x and y.
- How accurate is the result?
- Euler's Method is first-order accurate. Reducing the step size lowers the local error proportionally to h. For stiff or highly curved solutions, consider more advanced methods.