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