This is the solution to 
         y' = y*cos(x),  y(0) = 1
using Taylor series methods of orders 2, 3, and 4.
Observe the behaviour of the error, at fixed values of x,
as h is halved.

Stepsize h=0.2

                                Errors in Numerical Solution
   x          Y(x)          Order=2        Order=3        Order=4
 0.000     1.00000000      0.000E+00      0.000E+00      0.000E+00
 1.000     2.31977682     -0.162E-01     -0.197E-02     -0.346E-05
 2.000     2.48257773     -0.232E-01      0.246E-03      0.779E-04
 3.000     1.15156284      0.232E-02     -0.681E-03     -0.411E-04
 4.000     0.46916419     -0.118E-02     -0.513E-03      0.293E-04
 5.000     0.38330500     -0.222E-02     -0.177E-03      0.258E-04
 6.000     0.75622563      0.108E-04     -0.107E-03      0.301E-04
 7.000     1.92897080     -0.575E-02     -0.195E-02     -0.249E-04
 8.000     2.68950792     -0.276E-01     -0.139E-02      0.190E-03
 9.000     1.51001334      0.940E-03     -0.695E-03     -0.512E-04
10.000     0.58040966      0.431E-03     -0.845E-03      0.354E-04
  
 
Stepsize h=0.1

 0.000     1.00000000      0.000E+00      0.000E+00      0.000E+00
 1.000     2.31977682     -0.445E-02     -0.255E-03      0.736E-06
 2.000     2.48257773     -0.595E-02      0.248E-04      0.482E-05
 3.000     1.15156284      0.259E-03     -0.892E-04     -0.178E-05
 4.000     0.46916419     -0.431E-03     -0.602E-04      0.187E-05
 5.000     0.38330500     -0.626E-03     -0.190E-04      0.162E-05
 6.000     0.75622563     -0.739E-04     -0.878E-05      0.166E-05
 7.000     1.92897080     -0.191E-02     -0.251E-03     -0.203E-05
 8.000     2.68950792     -0.746E-02     -0.163E-03      0.110E-04
 9.000     1.51001334     -0.236E-03     -0.942E-04     -0.303E-05
10.000     0.58040966     -0.142E-03     -0.998E-04      0.209E-05
  
 
Stepsize h=0.05

 0.000     1.00000000      0.000E+00      0.000E+00      0.000E+00
 1.000     2.31977682     -0.116E-02     -0.325E-04      0.788E-07
 2.000     2.48257773     -0.151E-02      0.276E-05      0.304E-06
 3.000     1.15156284      0.250E-04     -0.114E-04     -0.874E-07
 4.000     0.46916419     -0.124E-03     -0.730E-05      0.119E-06
 5.000     0.38330500     -0.165E-03     -0.220E-05      0.102E-06
 6.000     0.75622563     -0.268E-04     -0.854E-06      0.967E-07
 7.000     1.92897080     -0.536E-03     -0.319E-04     -0.140E-06
 8.000     2.68950792     -0.193E-02     -0.200E-04      0.660E-06
 9.000     1.51001334     -0.117E-03     -0.123E-04     -0.184E-06
10.000     0.58040966     -0.653E-04     -0.122E-04      0.127E-06