Numerical Methods: Taylor Series

Question:

Obtain a numerical solution of the differential equation:

in the range 1.0(0.2)2.0 using the Taylor series method of order 4, given the initial conditions that

x = 1 when y = 4.

Answer:

Use initial conditions to calculate the all values at their initial point:

Use the Taylor series to find an approximation of y at x0+h:

At x = 1.6 the actual solution to the equation y'=3(1+x) - y is y = 5.348811636. Therefore, the percentage error of this approximation is:

Therefore, in this case, the Taylor series approximation of order 4 gives a very accurate answer. The percentage error will be dependent in the order of Taylor series approximation; the higher the order, the smaller the error.