BAB VPENYELESAIAN PERSAMAAN DIFERENSIAL ORDINER SIMULTANDENGAN RUNGE KUTTA

A. TUJUANAgar mahasiswa dapat menyelesaikan bentuk penyelesaian differensial ordiner simultan menggunakan penyelesaian numerik.B. DASAR TEORIJika dijumpai bentuk :

I.c : x = x0 ; y = y0 ; z = z0 Maka cara Range Kutta untuk mencari xi+1, yi+1,zi+1 berdasar harga xi,yi,zi adalah :K1 = f1 ( xi , yi, zi ) . L1 = = f2 ( xi , yi, zi ) .

K2 = f1 ( xi + / 2 , yi + k1 / 2 + l1/2 ) .L2 = f2 ( xi + / 2 , yi + k1 / 2 + l1/2 ) .

K3 = f1 (xi + / 2 , yi + k2 / 2 + l1/2 ) .L3 = f2 (xi + / 2 , yi + k2 / 2 + l1/2 ) .

K4 = f1 ( xi + , yi + k3 ,z1+l3) . L4 = f2 ( xi + , yi + k3 ,z1+l3) .

Dapat diperoleh :Xi+1= xi + Yi+1= yi + ( k1 + 2k2 + 2k3 + k4 ) / 6}Zi+1= zi + ( l1 + 2l2 + 2l3 + l4 ) / 6}

Algoritma1. Menentukan persamaan yang akan diselesaikan

2. Menentukan nilai X0,Y0,Z0,Xn,n,x3. Menghitung nilai K1 L1,K2 L2,K3 L3,K4 L4.4. Menghitung harga x,y,z baru

BAB 2x00,5

y01

z01,5

xn1,5

n10

x0,1

A. LATIHAN1.

TENTUKAN NILAI y,z PADA SAAT x = 1.5 DENGAN MENGGUNAKAN METODE RANGE KUTTA ?nx0y0z0k1l1k2l2k3l3k4l4yz

00,511,50,12500,24750,13790,25890,13830,25920,15200,27131,13821,7592

10,61,13821,75920,15200,27130,16640,28390,16690,28420,18210,29741,30502,0433

20,71,30502,04330,18210,29740,19820,31120,19880,31160,21580,32601,50372,3548

30,81,50372,35480,21580,32600,23380,34090,23450,34130,25350,35691,73802,6960

40,91,73802,69600,25340,35690,27340,37310,27420,37340,29530,39022,01203,0694

512,01203,06940,29530,39020,31750,40760,31840,40790,34180,42592,33013,4772

61,12,33013,47720,34180,42590,36640,44450,36730,44480,39320,46402,69723,9220

71,22,69723,92200,39320,46400,42030,48370,42130,48410,44990,50443,11834,4060

81,33,11834,40600,44990,50440,47970,52540,48080,52570,51220,54733,59884,9316

91,43,59884,93160,51220,54730,54500,56940,54610,56970,58050,59244,14465,5013

101,54,14465,50130,58040,59240,616290,61570,61750,61610,65500,64004,76186,1173

Maka untuk x =2 harga y =5.1344 dan z = 4.0399x01

y01

z00,1

xn2

n20

x0,05

2.

nx0y0z0k1l1k2l2k3l3k4l4yz

0110,10,06580,02660,06820,02820,06830,02820,07080,02991,06830,1282

11,051,06830,12820,07080,02990,07340,03170,07340,03180,07610,03371,14170,1600

21,11,14170,16000,07610,03370,07890,03570,07900,03570,08190,03781,22060,1957

31,151,22060,19570,08190,03780,08490,04010,08500,04010,08820,04251,30560,2359

41,21,30560,23590,08820,04250,09150,04510,09160,04520,09510,04791,39720,2810

51,251,39720,28100,09510,04790,09870,05080,09880,05080,10260,05391,49600,3318

61,31,49600,33180,10260,05390,10670,05730,10680,05740,11110,06091,60270,3892

71,351,60270,38920,11110,06090,11550,06480,11570,06490,12050,06911,71840,4541

81,41,71840,45410,12050,06910,12550,07350,12570,07370,13100,07861,84410,5278

91,451,84410,52780,13100,07860,13670,08390,13700,08410,14300,08991,98100,6119

101,51,98100,61190,14300,08990,14940,09630,14970,09660,15660,10362,13060,7084

111,552,13060,70840,15660,10360,16400,11130,16440,11170,17230,12032,29490,8200

121,62,29490,82000,17230,12020,18080,12970,18130,13030,19050,14102,47610,9502

131,652,47610,95020,19050,14090,20040,15280,20100,15360,21190,16712,67701,1037

141,72,67701,10370,21180,16710,22350,18230,22430,18350,23720,20092,90111,2870

151,752,90111,28700,23720,20090,25110,22080,25220,22250,26770,24573,15301,5092

161,83,15301,50920,26770,24560,28460,27220,28600,27480,30500,30653,43871,7835

171,853,43871,78350,30500,30640,32590,34330,32780,34730,35160,39213,76602,1301

181,93,76602,13010,35150,39190,37800,44480,38070,45140,41120,51744,14602,5804

191,954,14602,58040,41100,51710,44530,59640,44930,60790,48960,71024,59433,1865

2024,59433,18650,48940,70960,53530,83550,54150,85690,59701,02635,13444,0399

Maka untuk x =2 harga y =5.1344 dan z = 4.0399