This method is referred to as the modified Euler method.

ii) .

The resulting method is

(3.26)

which is known as the Improved Euler Method.

For , we can match (3.21) with (3.16) upto and including term if we satisfy the following set of equations

(3.27)

These are now four equations in six unknowns and there exists a two–parameter family of solutions. Thus there exist infinite family of three-stage Runge-kutta methods of order three and none of order more than three. Two particular solutions of (3.27) lead to well-known third order Runge–kutta methods

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The resulting method is:

(3.28)

This is known as Heun's third order method.