Runge–Kutta method

You can use this calculator to solve first-degree differential equation with a given initial value using the Runge-Kutta method AKA classic Runge-Kutta method (because there is a family of Runge-Kutta methods) or RK4 (because it is a fourth-order method).

To use this method, you should have differential equation in the form

and enter the right side of the equation f(x,y) in the y' field below.

You also need initial value as

and the point for which you want to approximate the value.

The last parameter of a method - a step size, is a step to compute the next approximation of a function curve.

Method details can be found below the calculator.

Runge–Kutta method

y'

Initial x

Initial y

Point of approximation

Step size

Calculation precision

Digits after the decimal point: 2

Calculate

Differential equation

$\displaystyle{y \prime = y}$

Approximate value of y

2.72

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Approximation

n	x{n}	y{n}	k1	k2	k3	k4	y{n+1}
0	0	1	0.1	0.11	0.11	0.11	1.11
1	0.1	1.11	0.11	0.12	0.12	0.12	1.22
2	0.2	1.22	0.12	0.13	0.13	0.13	1.35
3	0.30	1.35	0.13	0.14	0.14	0.15	1.49
4	0.4	1.49	0.15	0.16	0.16	0.16	1.65
5	0.5	1.65	0.16	0.17	0.17	0.18	1.82
6	0.6	1.82	0.18	0.19	0.19	0.20	2.01
7	0.7	2.01	0.20	0.21	0.21	0.22	2.23
8	0.80	2.23	0.22	0.23	0.23	0.25	2.46
9	0.90	2.46	0.25	0.26	0.26	0.27	2.72
10	1.00	2.72
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The Runge-Kutta method

Just like Euler method and Midpoint method, the Runge-Kutta method is a numerical method that starts from an initial point and then takes a short step forward to find the next solution point.

The formula to compute the next point is

where h is step size and

The local truncation error of RK4 is of order , giving a global truncation error of order .