Given the IVP of Eq. 6, and a time step *h*, and the solution *y*_{n} at the *n*th time step,
let's say that we wish to compute *y*_{n+1} in the following fashion:

k_{1} = hf(y_{n},t_{n}) |
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y_{n+1} = y_{n} + ak_{1} + bk_{2}, |
(12) |

where the constants , ,

Now, let's write down the Taylor series expansion of *y* in the neighborhood of *t*_{n} correct to
the *h*^{2} term i.e.,

(13) |

However, we know from the IVP (Eq. 6) that

(14) |

So from the above analysis, i.e., Eqs. 14 and 15, we get

(15) |

However, the term

(16) |

Now, substituting for

(17) |

Comparing the terms with identical coefficients in Eqs. 16 and 18 gives us the following system of equations to
determine the constants:

a+b=1 |
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(18) |

There are infinitely many choices of

k_{1} = hf(y_{n},t_{n}) |
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k_{2} = hf(y_{n}+k_{1}, t_{n} + h) |
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(19) |

In a similar fashion Runge-Kutta methods of higher order can be developed. One of the most widely used methods for the
solution of IVPs is the *fourth order Runge-Kutta* (RK4) technique. The LTE of this method is order *h*^{5}.
The method is given below.

k_{1} = hf(y_{n},t_{n}) |
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k_{2} = hf(y_{n}+k_{1}/2, t_{n} + h/2) |
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k_{4} = h(y_{n}+k_{3}, t_{n} + h) |
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y_{n+1} = y_{n} + (k_{1} + 2k_{2} + 2k_{3} + k_{4})/6. |
(20) |

Note that the RK methods are explicit techniques, hence they are only conditionally stable.