Taylor Series method

The principle of Taylor series method is to represent the solution of a deferential equation locally by a few terms of its Taylor series.

If the series is truncated after the first order, we get Euler's method presented above. The same equation as before is presented here to illustrate Taylor series method.

If the differential equation is differentiated several times with respect to t, the results are as follows:

If t and x(t) are known, these four formulas, applied in order, yield x', x'', x''', and x''''. Thus, it is possible from this work to use the first five terms in the Taylor series of the original equation.

This link will take you to an applet demonstrating Taylor series method.