persamaan diferensial dengan menggunakan metode numerik

1. Nama Anggota Kelompok:
Okta Dwi Rahmawati (12610079)
Ainus Sofiy (12610088)
Ruhmaa Mufida (12610101)
Contoh Soal.2 halaman 475 BAB 8 NUMERICAL METHOD
Diberikan persamaan dan nilai awal sebagai berikut : 푑푦 푡 푑푡 =−100푦 푡 +100푡+1 푦 0 =1 0≤푡≤1
Temukan solusinya dengan menggunakan pendekatan numerik (Metode Euler, Backward Euler, Runge- Kutta)
 Secara Analitik
푑푦 푑푡 = -100y + 100t + 1
푑푦 푑푡 + 100y = 100t + 1
P= 100
푃 푑푡 = 100t
Q = 100t + 1
Y(t) . 푒 푃 푑푡 = 푄 . 푒 푃 푑푡 dt + c
= (100푡+1) . 푒100푡 dt + c
= (100푡) 푒100푡 dt + 푒100푡 dt + c
= 푡푒100푡 − 푒100푡 dt + 푒100푡 100 + c
= 푡푒100푡 − 푒100푡 100 + 푒100푡 100 + c
= 푡푒100푡 + c
Solusi Umum
Y(0) = 0 + c = 1
c = 1
Solusi Khusus
푑푣 = 푒100푡 dt
v = 푒100푡 100
u = 100t
du = 100dt
Y(t) = t + c 푒−100푡
Y(t) = t + 푒−100푡

2.  Secara Numerik
 Metode Euler h=0,025 dan h=0,0166
Dengan menggunakan rumus: 푦푖+1=푦푖+푓 푡푖,푦푖 .ℎ
Dari contoh soal, diketahui 푦′ sebagai berikut: 푦′=−100푦+100푡+1 ,푦 0 =1
 H = 0.025
 푠푎푎푡 푡=0.025,푚푎푘푎
푓 0,1 =−100 1 +100 0 +1 =−100+0+1
=−99 dengan menggunakan metode euler didapat: 푦1=푦0+푓 푡0,푦0 .ℎ
=1+푓 0,1 0.025 =1−99 0.025 =1−2.475 =−1.475
 푠푎푎푡 푡=0.05,푚푎푘푎
푓 0.025,−1.475 =−100 −1.475 +100 0.025 +1 =147.5+2.5+1
=150.5 dengan menggunakan metode euler didapat: 푦2=푦1+푓 푡1,푦1 .ℎ
=−1.475+푓 0.025,−1.475 0.025 =−1.475+150.5 0.025 =−1.475+3.7625 =2.2875
 푠푎푎푡 푡=0.075,푚푎푘푎
푓 0.05,2.2875 =−100 2.2875 +100 0.05 +1 =−228.75+5+1
=−226.75 dengan menggunakan metode euler didapat: 푦3=푦2+푓 푡2,푦2 .ℎ
=2.2875+푓 0.05,2.2875 0.025 =2.2875−226.75 0.025 =2.2875−5.66875 =−3.38125
 푠푎푎푡 푡=0.1,푚푎푘푎
푓 0.075,−3.38125 =−100 −3.38125 +100 0.075 +1 =338.125+7.5+1
=346.625 dengan menggunakan metode euler didapat: 푦3=푦2+푓 푡2,푦2 .ℎ
=−3.38125+푓 0.075,−3.38125 0.025 =−3.38125+346.625 0.025

3. =−3.38125+8.665625 =5.284375
 Dan seterusnya sampai titik t=1 (terlampir)
 H = 0,0166
 푠푎푎푡 푡=0.0166,푚푎푘푎
푓 0,1 =−100 1 +100 0 +1 =−100+0+1
=−99 dengan menggunakan metode euler didapat: 푦1=푦0+푓 푡0,푦0 .ℎ
=1+푓 0,1 0.0166 =1−99 0.0166 =1−1.6434
=−0.6434
 푠푎푎푡 푡=0.0332,푚푎푘푎
푓 0.0332,−0.6434 =−100 −0.06434 +100 0.0332 +1 =64.34+3.32+1
=68.66 dengan menggunakan metode euler didapat: 푦2=푦1+푓 푡1,푦1 .ℎ
=−0.6434 +푓 0.0332,−0.6434 0.0166 =−0.6434+68.66 0.0166 =−0.6434+1.1397 =0.4963
 푠푎푎푡 푡=0.0498,푚푎푘푎
푓 0.0498,0.4963 =−100 0.4963 +100 0.0498 +1 =−49.63+4.98+1
=−43.65 dengan menggunakan metode euler didapat: 푦3=푦2+푓 푡2,푦2 .ℎ
=0.4963 +푓 0.0498,0.4963 0.0166 =0.4963−43.65 0.0166 =0.4963−0.7245 =−0.2282
 푠푎푎푡 푡=0.0664,푚푎푘푎
푓 0.0664,−0.2282 =−100 −0.2282 +100 0.0664 +1 =22.82+6.64+1
=30.46 dengan menggunakan metode euler didapat: 푦4=푦3+푓 푡3,푦3 .ℎ
=−0.2282+푓 0.0664,−0.2282 0.0166 =−0.2282+30.46 0.0166 =−0.2282+0.5056 =0.2774
 Dan seterusnya sampai titik t=1 (terlampir)
 Metode Runge-Kutta
Dengan menggunakan rumus: 푦푛+1=푦푛+ℎ 푘푛1+2푘푛2+2푘푛3+푘푛46

4. 푘푛1 = f(푡푛,푦푛)
푘푛2= f(푡푛+ 12 ℎ,푦푛+12 ℎ푘푛1 )
푘푛3= f(푡푛+ 12 ℎ,푦푛+12 ℎ푘푛2 )
푘푛4= f(푡푛+ℎ,푦푛+ℎ푘푛3 )
Dari contoh soal, diketahui 푦′ sebagai berikut: 푦′=−100푦+100푡+1 ,푦 0 =1
 H = 0,025
 saat t = 0
푘푛1 = f (0 ; 1)
= − 100 1 + 100 0 + 1= −99 푘푛2=푓 0,0125; −0,2375
= −100 −0,2375 +100 0,0125 +1= 26 푘푛3=푓 0,0125; 1,325
= −100 1,325 + 100 0,0125 +1= −130,25 푘푛4=푓 0,025; −2,25625
= −100 −2,25625 + 100 0,025 + 1=229,125 푦 0,025 =1+0,025 −99+2 26 + 2 −130,25 +229,125 6
= 0,6734375
 saat t = 0,025
푘푛1 = f (0,025 ; 0,6734375 )
= − 100 0,6734375 + 100 0,025 + 1= −63,84375 푘푛2=푓 0,0375; −0,124609375
= −100 −0,124609375 +100 0,0375 +1= 17,2109375 푘푛3=푓 0,0375; 0,8885742188
= −100 0,8885742188 + 100 0,0375 +1= −84,10742188 푘푛4=푓 0,05; −1,429248047
= −100 −1,429248047 + 100 0,05 + 1=148,9248047

5. 푦 0,05 =0,6734375 +0,025 −63,84375+2 17,2109375 + 2 −84,10742188 +148,9248047 6
= 0,470471191
 saat t = 0,05
푘푛1 = f (0,05 ; 0,470471191)
= − 100 0,470471191 + 100 0,05 + 1= −41,0471191 푘푛2=푓 0,0625; −0,042617798
= −100 −0,042617798 +100 0,0625 +1= 11,5117798 푘푛3=푓 0,0625; 0,6143684385
= −100 0,6143684385 + 100 0,0625 +1= −54,18684385 푘푛4=푓 0,075; −0,884199905
= −100 −0,884199905 + 100 0,075 + 1=96,9199905
푦 0,075 =0,470471191 +0,025 −41,0471191+2 11,5117798 + 2 −54,18684385 +96,9199905 6
= 0,347649288
 Dan seterusnya sampai titik t=1 (terlampir)
 H = 0,0333
 saat t=0
푘푛1 = f (0 ; 1)
= − 100 1 + 100 0 + 1= −99 푘푛2=푓 0,01665; −0,64835
= −100 −0,64835 +100 0,01665 +1=67,5 푘푛3=푓 0,01665; 2,123875
= −100 2,123875 + 100 0,01665 +1= −209,7225 푘푛4=푓 0,0333; −5,98375925
= −100 −5,98375925 + 100 0,0333 + 1=602,705925
푦 0,0333 =1+0,0333 −99+2 67,5 + 2 −209,7225 +602,705925 6
= 2,2168981338

6.  saat t = 0,0333
푘푛1 = f (0,0333 ; 2,2168981338 )
= − 100 2,2168981338 + 100 0,0333 + 1= −217,3598134 푘푛2=푓 0,04995; −1,402142759
= −100 −1,402142759 +100 0,04995 +1=146,2092759 푘푛3=푓 0,04995; 4,6512825775
= −100 4,6512825775 + 100 0,04995 +1= −459,1332578 푘푛4=푓 0,0666; −13,07223935
= −100 −13,07223935 + 100 0,0666 + 1=1314,883935
푦 0,0666 =2,2168981338 +0,0333 −217,3598134+2 146,2092759 + 2 −459,1332578 +1314,883935 6
= 4,8347008096
 saat t = 0,0666
푘푛1 = f (0,0666 ; 4,8347008096)
= − 100 4,8347008096 + 100 0,0666 + 1= −457,810081 푘푛2=푓 0,08325; −3,087537039
= −100 −3,087537039 +100 0,08325 +1=318,0787039 푘푛3=푓 0,08325; 10,13071123
= −100 10,13071123 + 100 0,08325 +1= −1003,746123 푘푛4=푓 0,0999; −28,5900450863
= −100 −28,5900450863 + 100 0,0666 + 1=2869,99450863
푦 0,0999 =4,834700809+0,0333 −457,810081+2 318,0787039 + 2 −1003,746123 +2869,994508636
= 10,5115160309365
 Dan seterusnya sampai titik t=1 (terlampir)
 Metode Backward
푦푖+1=푦푖+푓 푡푖,푦푖 .ℎ

7. 푦′=−100푦+100푡+1 푑푦 푑푡 =−100푦+100푡+1 푦푖+1−푦푖 Δ푡 =−100푦푖+1+100푡푖+1+1 푦푖+1Δ푡 +100푦푖+1= 푦푖 Δ푡 +100푡푖+1+1 푦푖+1 1+100Δ푡 Δ푡 = 푦푖 Δ푡 +100푡푖+1+1 푦푖+1= 푦푖 Δ푡 +100푡푖+1+1 .Δ푡 1+100Δ푡
Dari persamaan diatas, dengan Δ푡=0.1 maka,
 saat 푡1=0.1 , diperoleh
푦1= 10.1+100 0.1 +1 .0.11+100 0.1 = 10+10+1 0.1 1+10 = 2.111 =0.1909
 saat 푡1=0.2 , diperoleh
푦2= 0.19090.1+100 0.2 +1 .0.11+100 0.1 = 1.909+20+1 0.1 1+10 = 2.2911 =0.2082
 saat 푡1=0.3 , diperoleh
푦3= 0.20820.1+100 0.3 +1 .0.11+100 0.1 = 2.082+30+1 0.1 1+10 = 3.30811 =0.3
 Dan seterusnya sampai titik t=1 (terlampir)

8. Tabel perhitungan metode euler dengan h = 0,025
t
Y
Z
0,000000
1,000000
1,000000
0,025000
-1,475000
0,107085
0,050000
2,300000
0,056738
0,075000
-3,300000
0,075553
0,100000
5,162500
0,100045
0,125000
-7,468750
0,125004
0,150000
11,540625
0,150000
0,175000
-16,910938
0,175000
0,200000
25,828906
0,200000
0,225000
-38,218359
0,225000
0,250000
57,915039
0,250000
0,275000
-86,222559
0,275000
0,300000
130,046338
0,300000
0,325000
-194,294507
0,325000
0,350000
292,279260
0,350000
0,375000
-437,518890
0,375000
0,400000
657,240836
0,400000
0,425000
-984,836253
0,425000
0,450000
1478,341880
0,450000
0,475000
-2216,362820
0,475000
0,500000
3325,756730
0,500000
0,525000
-4987,360095
0,525000
0,550000
7482,377643
0,550000
0,575000
-11222,166464
0,575000
0,600000
16834,712196
0,600000
0,625000
-25250,543294
0,625000
0,650000
37877,402441
0,650000
0,675000
-56814,453662
0,675000
0,700000
85223,392992
0,700000
0,725000
-127833,314489
0,725000
0,750000
191751,809233
0,750000
0,775000
-287625,813849
0,775000
0,800000
431440,683274
0,800000
0,825000
-647158,999911
0,825000
0,850000
970740,587366
0,850000
0,875000
-1456108,731050
0,875000
0,900000
2184165,309075
0,900000
0,925000
-3276245,688612
0,925000
0,950000
4914370,870418
0,950000
0,975000
-7371553,905627
0,975000
1,000000
11057333,320940
1,000000

9. Grafik 1 Metode Euler h=0,025 (Ms. Excel)
Grafik 2 Metode Euler h=0,025 (Matlab)
Tabel perhitungan metode euler dengan h = 0,0166
t
Y
Z
0,000000
1,000000
1,000000
0,016600
-0,643400
0,206739
0,033200
0,468800
0,069353
0,049800
-0,237696
0,056674
0,066400
0,256147
0,067707
0,083000
-0,042233
0,083249
-10000000,000000
-5000000,000000
0,000000
5000000,000000
10000000,000000
15000000,000000
0,000000
0,200000
0,400000
0,600000
0,800000
1,000000
1,200000
GRAFIK SOLUSI ANALITIK DAN SOLUSI NUMERIK(METODE EULER) H=0,025
Y
Z

10. 0,099600
0,182254
0,099647
0,116200
0,061648
0,116209
0,132800
0,168804
0,132802
0,149400
0,125637
0,149400
0,166000
0,181683
0,166000
0,182600
0,172249
0,182600
0,199200
0,206032
0,199200
0,215800
0,211291
0,215800
0,232400
0,235376
0,232400
0,249000
0,247036
0,249000
0,265600
0,266896
0,265600
0,282200
0,281344
0,282200
0,298800
0,299365
0,298800
0,315400
0,315027
0,315400
0,332000
0,332246
0,332000
0,348600
0,348438
0,348600
0,365200
0,365307
0,365200
0,381800
0,381729
0,381800
0,398400
0,398447
0,398400
0,415000
0,414969
0,415000
0,431600
0,431620
0,431600
0,448200
0,448187
0,448200
0,464800
0,464809
0,464800
0,481400
0,481394
0,481400
0,498000
0,498004
0,498000
0,514600
0,514597
0,514600
0,531200
0,531202
0,531200
0,547800
0,547799
0,547800
0,564400
0,564401
0,564400
0,581000
0,581000
0,581000
0,597600
0,597600
0,597600
0,614200
0,614200
0,614200
0,630800
0,630800
0,630800
0,647400
0,647400
0,647400
0,664000
0,664000
0,664000
0,680600
0,680600
0,680600
0,697200
0,697200
0,697200
0,713800
0,713800
0,713800
0,730400
0,730400
0,730400
0,747000
0,747000
0,747000
0,763600
0,763600
0,763600
0,780200
0,780200
0,780200
0,796800
0,796800
0,796800
0,813400
0,813400
0,813400
0,830000
0,830000
0,830000
0,846600
0,846600
0,846600
0,863200
0,863200
0,863200
0,879800
0,879800
0,879800
0,896400
0,896400
0,896400

11. 0,913000
0,913000
0,913000
0,929600
0,929600
0,929600
0,946200
0,946200
0,946200
0,962800
0,962800
0,962800
0,979400
0,979400
0,979400
0,996000
0,996000
0,996000
1,012600
1,012600
1,012600
Grafik 3 Metode Euler h=0,0166 (Ms. Excel)
Grafik 4 Metode Euler h=0,0166 (Matlab)
Tabel untuk metode runge-kutta h= 0,0333
t
Y
K1
K2
K3
K4
Z
0
1
-99
67,5
-209,723
602,7059
1,000000
-1,000000
-0,500000
0,000000
0,500000
1,000000
1,500000
0,000000
0,200000
0,400000
0,600000
0,800000
1,000000
1,200000
GRAFIK SOLUSI ANALITIK DAN SOLUSI NUMERIK(METODE EULER ) h = 0,0166
Y
Z

12. 0,0333
2,216898
-217,3598134
146,2093
-459,133
1314,884
0,069093
0,0666
4,834701
-475,810081
318,0787
-1003,75
2869,995
0,067881
0,0999
10,51152
-1040,161603
693,3725
-2192,96
6265,731
0,099946
0,1332
22,86799
-2272,478533
1512,863
-4789,73
13680,66
0,133202
0,1665
49,81013
-4963,363482
3302,302
-10460
29871,87
0,166500
0,1998
108,6015
-10839,17483
7209,716
-22841,7
65226,97
0,199800
0,2331
236,939
-23669,58554
15741,94
-49878,2
142428,3
0,233100
0,2664
517,1369
-51686,04641
34372,89
-108915
311005
0,266400
0,2997
1128,937
-112862,7381
75055,39
-237828
679108,8
0,299700
0,333
2464,823
-246448,0478
163889,6
-519323
1482900
0,333000
0,3663
5381,823
-538144,6809
357867,9
-1133993
3238055
0,366300
0,3996
11751,34
-1175092,904
781438,4
-2476186
7070611
0,399600
0,4329
25659,76
-2565931,857
1706346
-5406997
15439371
0,432900
0,4662
56030,13
-5602965,198
3725974
-1,2E+07
33713381
0,466200
0,4995
122346,8
-12234625,53
8136028
-2,6E+07
73616474
0,499500
0,5328
267155,6
-26715506,66
17765814
-5,6E+07
1,61E+08
0,532800
0,5661
583359,9
-58335931,68
38793396
-1,2E+08
3,51E+08
0,566100
0,5994
1273823
-127382232,7
84709186
-2,7E+08
7,66E+08
0,599400
0,6327
2781517
-278151606,8
1,85E+08
-5,9E+08
1,67E+09
0,632700
0,666
6073714
-607371330,8
4,04E+08
-1,3E+09
3,65E+09
0,666000
0,6993
13262550
-1326254906
8,82E+08
-2,8E+09
7,98E+09
0,699300
0,7326
28960078
-2896007738
1,93E+09
-6,1E+09
1,74E+10
0,732600
0,7659
63237172
-6323717093
4,21E+09
-1,3E+10
3,81E+10
0,765900
0,7992
1,38E+08
-13808456843
9,18E+09
-2,9E+10
8,31E+10
0,799200
0,8325
3,02E+08
-30152120593
2,01E+10
-6,4E+10
1,81E+11
0,832500
0,8658
6,58E+08
-65840114257
4,38E+10
-1,4E+11
3,96E+11
0,865800
0,8991
1,44E+09
-1,43768E+11
9,56E+10
-3E+11
8,65E+11
0,899100
0,9324
3,14E+09
-3,13932E+11
2,09E+11
-6,6E+11
1,89E+12
0,932400
0,9657
6,86E+09
-6,85502E+11
4,56E+11
-1,4E+12
4,12E+12
0,965700
0,999
1,5E+10
-1,49686E+12
9,95E+11
-3,2E+12
9,01E+12
0,999000
Grafik 5 Metode Runge-Kutta h=0,0333 (Ms.Excel)
-2E+09
0
2E+09
4E+09
6E+09
8E+09
1E+10
1,2E+10
1,4E+10
1,6E+10
0
0,2
0,4
0,6
0,8
1
1,2
GRAFIK SOLUSI ANALITIK DAN SOLUSI NUMERIK(METODE Runge-Kutta ) h = 0,0333
Y
Z

13. Grafik 6 Metode Runge-Kutta h=0,0333 (Matlab)
Tabel metode Runge-Kutta h=0,025
t
K1
K2
K3
K4
Y
Z
0,000000
-99,000000
26,000000
-130,250000
65,062500
1,000000
1,000000
0,025000
76,234375
-17,808594
99,745117
-47,197021
-0,727344
0,107085
0,050000
-59,982971
16,245743
-79,040150
40,067216
0,659830
0,056738
0,075000
45,904732
-10,226183
59,937460
-27,767094
-0,374047
0,075553
0,100000
-36,406412
10,351603
-48,095916
24,963483
0,474064
0,100045
0,125000
27,577641
-5,644410
35,883154
-16,026301
-0,140776
0,125004
0,150000
-22,159963
6,789991
-29,397451
15,836851
0,381600
0,150000
0,175000
16,503253
-2,875813
21,348019
-8,931771
0,019967
0,175000
0,200000
-13,551356
4,637839
-18,098655
10,321963
0,345514
0,200000
0,225000
9,811406
-1,202851
12,564970
-4,644807
0,136886
0,225000
0,250000
-8,349491
3,337373
-11,271207
6,989518
0,343495
0,250000
0,275000
5,767769
-0,191942
7,257696
-2,054352
0,227322
0,275000
0,300000
-5,206195
2,551549
-7,145631
4,975844
0,362062
0,300000
0,325000
3,324347
0,418913
4,050705
-0,489035
0,301757
0,325000
0,350000
-3,306817
2,076704
-4,652697
3,759054
0,393068
0,350000
0,375000
1,847877
0,788031
2,112839
0,456829
0,366521
0,375000
0,400000
-2,159092
1,789773
-3,146308
3,023793
0,431591
0,400000
0,425000
0,955700
1,011075
0,941857
1,028379
0,425443
0,425000
0,450000
-1,465564
1,616391
-2,236053
2,579502
0,474656
0,450000
0,475000
0,416591
1,145852
0,234275
1,373747
0,480834
0,475000
0,500000
-1,046490
1,511623
-1,686019
2,311033
0,520465
0,500000
0,525000
0,090827
1,227293
-0,193290
1,582439
0,534092
0,525000
0,550000
-0,793260
1,448315
-1,353653
2,148807
0,567933
0,550000
0,575000
-0,106021
1,276505
-0,451652
1,708545
0,586060
0,575000
0,600000
-0,640242
1,410060
-1,152817
2,050780
0,616402
0,600000

14. 0,625000
-0,224968
1,306242
-0,607771
1,784745
0,637250
0,625000
0,650000
-0,547778
1,386945
-1,031459
1,991546
0,665478
0,650000
0,675000
-0,296844
1,324211
-0,702108
1,830791
0,687968
0,675000
0,700000
-0,491906
1,372977
-0,958127
1,955753
0,714919
0,700000
0,725000
-0,340276
1,335069
-0,759112
1,858614
0,738403
0,725000
0,750000
-0,458145
1,364536
-0,913815
1,934124
0,764581
0,750000
0,775000
-0,366520
1,341630
-0,793558
1,875427
0,788665
0,775000
0,800000
-0,437744
1,359436
-0,887039
1,921055
0,814377
0,800000
0,825000
-0,382379
1,345595
-0,814372
1,885586
0,838824
0,825000
0,850000
-0,425417
1,356354
-0,870859
1,913158
0,864254
0,850000
0,875000
-0,391961
1,347990
-0,826949
1,891725
0,888920
0,875000
0,900000
-0,417968
1,354492
-0,861082
1,908385
0,914180
0,900000
0,925000
-0,397752
1,349438
-0,834549
1,895435
0,938978
0,925000
0,950000
-0,413466
1,353367
-0,855175
1,905502
0,964135
0,950000
0,975000
-0,401251
1,350313
-0,839142
1,897676
0,989013
0,975000
1,000000
-0,410747
1,352687
-0,851605
1,903759
1,014107
1,000000
Grafik 7 Metode Runge-Kutta h=0,025 (Ms.Excel)
Grafik 8 Metode Runge-Kutta h=0,025 (Matlab)
-1,000000
-0,800000
-0,600000
-0,400000
-0,200000
0,000000
0,200000
0,400000
0,600000
0,800000
1,000000
1,200000
0,000000
0,200000
0,400000
0,600000
0,800000
1,000000
1,200000
GRAFIK SOLUSI UMERIK DAN SOLUSI ANALITIK (METODE RUNGE KUTTA) H= 0,025
Y
Z

15. Tabel untuk metode backward euler h=0,1
t
Y
Z
0,000000
0,100000
1,000000
0,100000
0,109091
0,100045
0,200000
0,200826
0,200000
0,300000
0,300075
0,300000
0,400000
0,400007
0,400000
0,500000
0,500001
0,500000
0,600000
0,600000
0,600000
0,700000
0,700000
0,700000
0,800000
0,800000
0,800000
0,900000
0,900000
0,900000
1,000000
1,000000
1,000000
Grafik 9 Metode Backward Euler h=0,1 (Ms.Excel)
Grafik 10 Metode Backward Euler h=0,1 (Matlab)
0,000000
0,200000
0,400000
0,600000
0,800000
1,000000
1,200000
0,000000
0,200000
0,400000
0,600000
0,800000
1,000000
1,200000
GRAFIK SOLUSI ANALITIK DAN SOLUSI NUMERIK(METODE BACKWARD EULER) H=0,1
Y
Z

16. GRAFIK Z(NILAI EXACT)
Grafik 11 Analitik (Ms.Excel)
Keterangan
 Y pada tabel dan grafik mewakili data dari nilai hampiran (numerik)
 Z pada tabel dan grafik mewakili data dari nilai eksak (analitik)
0,000000
0,200000
0,400000
0,600000
0,800000
1,000000
1,200000
0,000000
0,200000
0,400000
0,600000
0,800000
1,000000
1,200000
Z
Z