Runge–Kutta Method Calculator

y′ = f(x, y)

Allowed: x, y, + − * / ( ) ^ and functions like sin, exp, log.

Method

RK45 adapts step size to meet a tolerance.

Step size h

For RK45, this is the initial step size.

x₀

y₀

x_end

Tolerance (RK45)

Used only when RK45 is selected.

Max steps

Stops safely if the target is not reached.

Decimals

Controls displayed rounding and downloads.

Reset

Example data

Try this sample initial value problem

This example uses y′ = x + y with y(0)=1 on [0, 1].

Equation y′	x₀	y₀	x_end	h	Suggested method
x + y	0	1	1	0.1	RK4 (Classic)
sin(x) - y	0	0	6.28	0.05	RK45 (Adaptive)

Formula used

Runge–Kutta IVP model

We solve y′ = f(x, y) with y(x₀)=y₀ using step updates.

Classic RK4 update

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), xn+1 = xn + h

Adaptive RK45 idea

RK45 computes a 5th-order and 4th-order estimate at each step. Their difference estimates local error, then the step size is adjusted to meet your tolerance.

How to use this calculator

Enter y′ = f(x, y) using allowed operators and functions.
Provide x₀, y₀, and the target x_end.
Choose a method. RK4 is a strong default. Use RK45 for accuracy control.
Set h and optional tolerance for RK45.
Press Submit. Your result appears above the form.

Notes

If values explode or become NaN, reduce step size or use RK45.
Solve backwards by setting x_end < x₀; the step direction adjusts automatically.
Expressions using ^ are treated as exponentiation.

Runge–Kutta IVP model

Classic RK4 update

Adaptive RK45 idea

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