Suppose we have ODE then

**Formula**:

Where

and

**Other variation name**: Improve Euler method, Heun's method, Midpoint method

**Error term**: , correct up to the second order term in Taylor series expansion.

The Taylor series expansion is

The computation is using spreadsheet that can be downloaded here

**Example: **

, set , with initial condition . The few first results and the graph of solution are given below.

Note that after , the solution is not correct. See Comparison.

Expanding the order Runge-Kutta formula, we have

Second term of the right hand side [ ] is the estimated range difference. The estimated range difference is computed based on a half of the slope at the *beginning * of the interval and the slope in the *middle * of the interval , as illustrated in the figure below

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See also:Numerical Excel tutorial, Dynamical System tutorial, Kardi Teknomo's Tutorial

**Preferable reference for this tutorial is**

Teknomo, Kardi (2015) Solving Ordinary Differential Equation (ODE). http:\\people.revoledu.com\kardi\ tutorial\ODE\