Euler's method is a first-order numerical procedure for solving ordinary differential equations with given initial values. Our calculator compares Euler, Improved Euler (Heun's), and Runge-Kutta methods with step-by-step solutions and interactive visualizations for accurate mathematical analysis.
| Iteration | x | y | Slope f(x,y) | Next y | ||
|---|---|---|---|---|---|---|
| Calculate a solution to see results | ||||||
Euler's method approximates solutions to ordinary differential equations by using the slope at each point to predict the next point. The formula is: y₁ = y₀ + h × f(x₀, y₀)
Smaller step sizes provide more accurate results but require more calculations. Our calculator lets you adjust step size dynamically to see the trade-off between accuracy and computational efficiency.
| Method | Order of Accuracy | Computational Cost | Best Used For |
|---|---|---|---|
| Euler's Method | O(h) | Low | Quick estimates, simple problems |
| Improved Euler (Heun's) | O(h²) | Medium | Better accuracy with moderate cost |
| Runge-Kutta 4th Order | O(h⁴) | High | High precision requirements |
Use our calculator above to compare these methods side-by-side with your own differential equations.
dy/dx = 0.1*y
Models exponential population growth where the growth rate is proportional to current population.
dy/dx = -0.2*(y-20)
Temperature cooling model where object cools toward ambient temperature of 20°C.
dy/dx = cos(x)
Basic oscillation pattern commonly found in physics and engineering systems.
Developed by mathematics educators and numerical analysis specialists, our Euler's Method Calculator provides accurate, step-by-step solutions for ordinary differential equations. Trusted by students, engineers, and researchers worldwide for learning and problem-solving.