What is the Runge-Kutta method?

Runge-Kutta methods are iterative numerical procedures for solving ordinary differential equations. The classical RK4 method evaluates the derivative four times per step to get high accuracy.

Which ODEs can I use this with?

This tool is designed for first-order ODEs of the form dy/dx = f(x, y) with a given initial condition y(x0) = y0.

RK4 is a 4th-order method: the local truncation error is O(h^5) and the global error is O(h^4). Smaller step sizes h give better accuracy but require more computation.

Runge-Kutta (RK4) Method Calculator

Numerically solve a first-order ODE \( \frac{dy}{dx} = f(x, y) \) using the classical Runge–Kutta method of order 4 (RK4). We show every intermediate slope (k1–k4) and the resulting y values.

1. Problem setup

Use JavaScript/Math syntax: x + y, x*y, Math.sin(x), y - x*x. We already expose common Math functions via with(Math){...}.

dy/dx = f(x, y)

Number of steps

x₀

y(x₀)

Step size h

2. Solution steps

No computation yet. Fill the form and click “Run RK4”.

Runge–Kutta 4th order formula

Given \(\frac{dy}{dx} = f(x, y)\) and a current point \((x_n, y_n)\) with step \(h\):


                k₁ = f(xₙ, yₙ) 

                k₂ = f(xₙ + h/2, yₙ + h·k₁/2) 

                k₃ = f(xₙ + h/2, yₙ + h·k₂/2) 

                k₄ = f(xₙ + h,   yₙ + h·k₃) 

                yₙ₊₁ = yₙ + (h/6)(k₁ + 2k₂ + 2k₃ + k₄)

This method is popular because it strikes an excellent balance between accuracy and computational cost.

Tips

If the solution diverges, try a smaller step size \(h\).
For stiff ODEs, RK4 might not be the best choice.
Always check your function syntax if you get NaN results.