What is the Runge-Kutta RK4 method?

RK4 is a classical numerical algorithm for solving ordinary differential equations. It evaluates the derivative f(x,y) at four points within each step and combines them in a weighted average, achieving 4th-order accuracy (local error O(h^5), global error O(h^4)).

What expressions can I use for f(x,y)?

Expressions are parsed by math.js. Use x and y as variables. Supported: +, -, *, /, ^ (power), parentheses, and functions like sin, cos, tan, exp, log, sqrt, abs, pi, e. Example: x*y + sin(x).

How do I choose a good step size h?

Smaller h gives more accuracy but more computation. For smooth ODEs, h = 0.1 is usually accurate enough. If the solution changes rapidly or you need high precision, try h = 0.01. A good rule is to halve h and check whether the results change significantly.

Why does the solution blow up or show NaN?

Some ODEs have solutions that grow to infinity in finite time. The solver stops early and warns you when |y| exceeds 10^9. Try reducing the number of steps or using a smaller x range to stay before the singularity.

Can I solve second-order or higher-order ODEs?

Not directly. Convert a second-order ODE into a system of two first-order ODEs by substituting z = dy/dx, then solve each equation separately. This tool handles single first-order ODEs only.

Runge-Kutta Calculator (RK4)

Name: Runge-Kutta Calculator (RK4)
Availability: InStock
Author: Nham Vu

Numerically solve any first-order ODE dy/dx = f(x,y) using the classical 4th-order Runge-Kutta method. Enter f(x,y), initial conditions, step size, and number of steps to get a table of solution values.

Presets:

ODE: dy/dx = f(x, y)

f(x, y) expression

Use x and y as variables. Supports sin, cos, exp, log, sqrt, ^, etc.

x₀ — initial x

y₀ — initial y

Step size h

Number of steps n

Solution Table

Step	x	y
Press "Solve with RK4" to see results.

Solution Curve y(x)

RK4 Formula Reference

k1 = h · f(x_n, y_n)

k2 = h · f(x_n + h/2, y_n + k1/2)

k3 = h · f(x_n + h/2, y_n + k2/2)

k4 = h · f(x_n + h, y_n + k3)

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

Summary

How it works

The 4th-order Runge-Kutta method advances the solution from (xn, yn) to (xn+h, yn+1) using four slope estimates: k1 = h·f(xn, yn), k2 = h·f(xn+h/2, yn+k1/2), k3 = h·f(xn+h/2, yn+k2/2), k4 = h·f(xn+h, yn+k3), then yn+1 = yn + (k1 + 2k2 + 2k3 + k4)/6. This weighted average achieves 4th-order accuracy (global error O(h^4)). Expressions are evaluated by math.js, allowing any formula using x, y, standard operators, and built-in functions.

Use cases

Solve motion equations such as dy/dx = -y + sin(x) numerically.
Verify analytical solutions of first-order ODEs against numerical approximations.
Explore how initial conditions affect the shape of solution curves.
Compute population growth models dy/dx = r·y·(1 - y/K) step by step.
Approximate solutions for ODEs with no closed-form antiderivative.

Frequently Asked Questions

Last updated: 2026-05-23 · Reviewed by Nham Vu