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2.2.1 Taylor Series Methods     continued...

To obtain an approximate solution of order p to the IVP (17) on , let and generate the sequences

where and .

The Taylor method of order p=1 is known as Euler's method:

To illustrate it, we approximate the solution to the IVP,

at with step . Here , so (29) simplifies to

Starting with and , we compute, truncating results to four decimal places:

The exact solution is , so , correct to four decimal places, and the magnitude of the true or global error in our approximation is . The approximate and exact solutions are represented graphically below in Figure 2, where the approximating values , and have been joined by straight line segments.

Figure 2: Plot of the Euler solution and exact solutions.