Problem

Given the IVP

y⁽¹⁾(t) = f( t, y(t) )
y(a) = y₀

approximate y(t) with an error less than ε_abs.

Assumptions

The function f(t, y) should be continuous in both variables.

We will use a variation of the Runge-Kutta-Fehlberg method to find two approximations of the value y(t₁) to find an optimal value of h.

Set h = t₁ − t₀. Let y₁ be the approximation of y(t₁).

Define

K₀ = f( y₀, t₀ )
K₁ = f( y₀ + ½ h K₀, t₀ + ½ h )
K₂ = f( y₀ + ½ h K₁, t₀ + ½ h )
K₃ = f( y₀ + h K₂, t₀ + h )

Then let