Learning Guide Learning Guide

6.2 Ordinary Differential Equations • 247Example: Taylor SeriesIn its general form, a series method solution to an ODE requires theforming of a Taylor series about t = 0 for some function f(t). Thus, youmust be able to obtain and manipulate the higher order derivatives ofyour function, f ′ (t), f ′′ (t), f ′′′ (t), and so on.Once you have obtained the derivatives, you substitute them into theTaylor series representation of f(t).> taylor(f(t), t);f(0) + D(f)(0) t + 1 2 (D(2) )(f)(0) t 2 + 1 6 (D(3) )(f)(0) t 3 +124 (D(4) )(f)(0) t 4 + 1120 (D(5) )(f)(0) t 5 + O(t 6 )As an example, consider Newton’s Law of Cooling:dθdt = − 1 (θ − 20), θ(0) = 100.10Using the D operator, you can easily enter the above equation intoMaple.> eq := D(theta) = -1/10*(theta-20);eq := D(θ) = − 110 θ + 2> ini := theta(0)=100;ini := θ(0) = 100The first step is to obtain the required number of higher derivatives.Determine this number from the order of your Taylor series. If you usethe default value of Order that Maple provides,> Order;6then you must generate derivatives up to order> dev_order := Order - 1;