Maple 9 Learning Guide - Maplesoft

7.2 Ordinary Differential Equations • 255You can specify which algorithm dsolve(..., type=numeric) useswhen solving your differential equation. Refer to the ?dsolve,numerichelp page.Example: 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, substitute them into the Taylorseries 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, enter the above equation into Maple.> eq := D(theta) = -1/10*(theta-20);eq := D(θ) = − 110 θ + 2> ini := theta(0)=100;ini := θ(0) = 100First obtain the required number of higher derivatives. Determine thisnumber from the order of your Taylor series. If you use the default valueof Order that Maple provides,> Order;6then you must generate derivatives up to order