Initial Value Problems - Numerical Methods in Engineering with MATLAB - page 257

d(3,1) = d(2,2);
d(3,2) = -0.001*y(2) - 0.01*x + 0.1;
d(4,1) = d(3,2);
d(4,2) = 0.0001*y(2) + 0.001*x - 0.01;

Here is the solution:

>> [x,y] = taylor(@fex7

_

2, 0, [0 1], 2, 0.25);

>> printSol(x,y,1)
x y1 y2

0.0000e+000 0.0000e+000 1.0000e+000

2.5000e-001 2.4431e-001 9.4432e-001

5.0000e-001 4.6713e-001 8.2829e-001

7.5000e-001 6.5355e-001 6.5339e-001

1.0000e+000 7.8904e-001 4.2110e-001

1.2500e+000 8.5943e-001 1.3281e-001

1.5000e+000 8.5090e-001 -2.1009e-001

1.7500e+000 7.4995e-001 -6.0625e-001

2.0000e+000 5.4345e-001 -1.0543e+000

The analytical solution of the problemis

5 x 2 990( e − 0 . 1 x

y = 100 x − + − 1)

543 45 and y (2) fromwhich weobtain y (2) = 0 . =− 1 . 0543, which agree with the nu-

merical solution.
The main drawback of the Taylor series methodisthat it requires repeateddiffer-
entiation of the dependent variables. These expressions may become very long and
thuserror-prone and tedioustocompute. Moreover, there is the extra workofcoding
each of the derivatives.

7.3 Runge-Kutta Methods

The aim of Runge-Kuttamethods istoeliminate the need forrepeateddifferentiation
of the differentialequations.Since nosuch differentiationis involvedinthe first-order
Taylor series integration formula

y ( x ) h y ( x + h ) = y ( x ) + = y ( x ) + F ( x , y ) h (7.8)

itcan beconsideredas the first-order Runge-Kuttamethod; it is also called Euler's
method .Duetoexcessivetruncation error,this methodis rarelyusedinpractice.