ODE Solver: Euler & Improved Euler

This was made for MA331 class Differential Equations in Washburn University. No login required.

Euler's method uses the slope at the current point to step forward: y_n+1 = y_n + h · f(x_n, y_n). Improved Euler (Heun's method) uses the average of two slopes for better accuracy: k₁ = f(x_n, y_n), k₂ = f(x_n + h, y_n + h·k₁), and y_n+1 = y_n + (h/2)(k₁ + k₂).

1. Differential equation

dy/dx = f(x, y). Type or use the keypad below. x and y are numbers (current point); don't write x(...) or y(...).

dy/dx =

Note: Trig is in radians. Use √ for square root (symbol, not "sqrt"). Use a/b for fractions (numerator over denominator).

2. Initial conditions & step

Initial x (x₀) Initial y (y₀)

Step size (h)

Final x

Results

Euler's method

n	x	yₙ	f(xₙ,yₙ)	yₙ₊₁

Improved Euler (Heun)

n	x	yₙ	f(xₙ,yₙ)	k₁	k₂	yₙ₊₁

Graph

Both methods on the same plot. Drag to pan, scroll to zoom.