Runge-Kutta Method Output:
Simple Harmonic Oscillator
(Matlab)

>> [x,y]=ode45(f,[0 .5],1)

x =

         0
    0.0125
    0.0250
    0.0375
    0.0500
    0.0625
    0.0750
    0.0875
    0.1000
    0.1125
    0.1250
    0.1375
    0.1500
    0.1625
    0.1750
    0.1875
    0.2000
    0.2125
    0.2250
    0.2375
    0.2500
    0.2625
    0.2750
    0.2875
    0.3000
    0.3125
    0.3250
    0.3375
    0.3500
    0.3625
    0.3750
    0.3875
    0.4000
    0.4125
    0.4250
    0.4375
    0.4500
    0.4625
    0.4750
    0.4875
    0.5000

y =

    1.0000
    1.0001
    1.0003
    1.0007
    1.0013
    1.0020
    1.0028
    1.0038
    1.0050
    1.0063
    1.0078
    1.0095
    1.0113
    1.0133
    1.0154
    1.0177
    1.0201
    1.0227
    1.0255
    1.0285
    1.0316
    1.0348
    1.0383
    1.0419
    1.0457
    1.0496
    1.0537
    1.0580
    1.0624
    1.0671
    1.0719
    1.0769
    1.0820
    1.0873
    1.0929
    1.0985
    1.1044
    1.1105
    1.1167
    1.1231
    1.1297

Error= 14.16%

Euler's Method Output:
Simple Harmonic Oscillator
(Matlab)

>> [x,y]=euler(f,0,1,pi,100)

x =

         0
    0.0100
    0.0200
    0.0300
    0.0400
    0.0500
    0.0600
    0.0700
    0.0800
    0.0900
    0.1000
    0.1100
    0.1200
    0.1300
    0.1400
    0.1500
    0.1600
    0.1700
    0.1800
    0.1900
    0.2000
    0.2100
    0.2200
    0.2300
    0.2400
    0.2500
    0.2600
    0.2700
    0.2800
    0.2900
    0.3000
    0.3100
    0.3200
    0.3300
    0.3400
    0.3500
    0.3600
    0.3700
    0.3800
    0.3900
    0.4000
    0.4100
    0.4200
    0.4300
    0.4400
    0.4500
    0.4600
    0.4700
    0.4800
    0.4900
    0.5000
    0.5100
    0.5200
    0.5300
    0.5400
    0.5500
    0.5600
    0.5700
    0.5800
    0.5900
    0.6000
    0.6100
    0.6200
    0.6300
    0.6400
    0.6500
    0.6600
    0.6700
    0.6800
    0.6900
    0.7000
    0.7100
    0.7200
    0.7300
    0.7400
    0.7500
    0.7600
    0.7700
    0.7800
    0.7900
    0.8000
    0.8100
    0.8200
    0.8300
    0.8400
    0.8500
    0.8600
    0.8700
    0.8800
    0.8900
    0.9000
    0.9100
    0.9200
    0.9300
    0.9400
    0.9500
    0.9600
    0.9700
    0.9800
    0.9900
    1.0000

y =

    3.1416
    3.1416
    3.1419
    3.1425
    3.1435
    3.1447
    3.1463
    3.1482
    3.1504
    3.1529
    3.1557
    3.1588
    3.1622
    3.1659
    3.1699
    3.1742
    3.1787
    3.1836
    3.1888
    3.1942
    3.1999
    3.2059
    3.2121
    3.2186
    3.2253
    3.2323
    3.2396
    3.2470
    3.2547
    3.2626
    3.2707
    3.2790
    3.2875
    3.2962
    3.3051
    3.3141
    3.3233
    3.3326
    3.3420
    3.3516
    3.3612
    3.3710
    3.3808
    3.3907
    3.4006
    3.4106
    3.4206
    3.4306
    3.4406
    3.4505
    3.4605
    3.4703
    3.4801
    3.4898
    3.4995
    3.5090
    3.5183
    3.5275
    3.5366
    3.5454
    3.5541
    3.5626
    3.5708
    3.5788
    3.5866
    3.5941
    3.6013
    3.6082
    3.6148
    3.6211
    3.6271
    3.6328
    3.6381
    3.6431
    3.6478
    3.6520
    3.6560
    3.6595
    3.6627
    3.6655
    3.6679
    3.6700
    3.6717
    3.6730
    3.6739
    3.6745
    3.6746
    3.6745
    3.6739
    3.6730
    3.6717
    3.6701
    3.6681
    3.6658
    3.6632
    3.6602
    3.6569
    3.6533
    3.6494
    3.6452
    3.6407

Error = 11.5%