ODE Euler Method Solver
Numerically solve first-order ODEs using Euler's method.
Choose a preset dy/dx function, set initial conditions, and see a step-by-step solution table.
Euler’s Method for ODEs
Euler’s method is the simplest numerical technique for solving ordinary differential equations (ODEs). Given an initial value problem: dy/dx = f(x, y), y(x₀) = y₀
The method steps forward using: y_{n+1} = y_n + h · f(x_n, y_n)
Where h is the step size and f(x, y) is the derivative function.
Algorithm
- Start at (x₀, y₀)
- Compute slope: k = f(x_n, y_n)
- Step forward: x_{n+1} = x_n + h, y_{n+1} = y_n + h·k
- Repeat for the desired number of steps
Accuracy
Euler’s method has first-order accuracy: the global error is O(h). Halving the step size roughly halves the error.
For better accuracy, use:
- Improved Euler (Heun’s method): averages start and end slopes
- RK4 (Runge-Kutta 4th order): gold standard, O(h⁴) accuracy
Example: dy/dx = y, y(0) = 1
The exact solution is y = eˣ.
| x | y (Euler, h=0.1) | y (Exact) | Error |
|---|---|---|---|
| 0.0 | 1.0000 | 1.0000 | 0.0000 |
| 0.1 | 1.1000 | 1.1052 | 0.0052 |
| 0.2 | 1.2100 | 1.2214 | 0.0114 |
| 0.5 | 1.6105 | 1.6487 | 0.0382 |
Error grows with each step — smaller h gives better results.
When To Use
Euler’s method is excellent for:
- Teaching the concept of numerical ODE solving
- Quick estimates when high precision isn’t needed
- Simple problems where analytical solutions are hard to find
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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