082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

20.8.1 Higherorderequations
20.8 Runge--Kutta method of order 4 625
Thetechniquesdiscussedforthesolutionofsinglefirst-orderequationsgeneralizereadilytohigherorderequationssuchasthosedescribedinSection20.3andSection20.4.It
isobviousthatacomputersolutionisessentialandthereareawidevarietyofcomputer
packages available tosolve such equations.
EXERCISES20.8
1 Findy(0.4)ify ′ = (x +y) 2 andy(0) = 1usingthe
Runge--Kutta method oforder 4. Take (a) h = 0.2
and(b) h = 0.1.
2 Repeat Question 1 in Exercises20.6 usingthe
fourth-orderRunge--Kutta method.
3 Repeat Question 2 in Exercises20.6 usingthe
fourth-orderRunge--Kutta method.
4 Repeat Question 3 in Exercises20.6 usingthe
fourth-orderRunge--Kutta method.
Solutions
1 (a)
2
x i
y i
0 1.0000
0.2 1.3085
0.4 2.0640
(b)
x i
y i
0 1.0000
0.1 1.1230
0.2 1.3085
0.3 1.5958
0.4 2.0649
x i y i y i y
(h = 0.5) (h = 0.25) (exact)
2.00 1.0000 1.0000 1.0000
2.25 -- 1.3900 1.3901
2.50 1.8078 1.8079 1.8080
2.75 -- 2.2507 2.2509
3.00 2.7163 2.7164 2.7165
4
x i y i y
(h = 0.1) (exact)
0.0 0.0000 0.0000
0.1 0.0052 0.0052
0.2 0.0214 0.0214
0.3 0.0499 0.0499
0.4 0.0918 0.0918
0.5 0.1487 0.1487
t i v i v
(h = 0.005) (exact)
0 0.0000 0.0000
0.005 0.2675 0.2673
0.010 0.3959 0.3954
3
x i y i y
(h = 0.25) (exact)
0.00 0.0000 0.0000
0.25 0.0340 0.0340
0.50 0.1487 0.1487
t i v i v
(h = 0.002) (exact)
0 0.0000 0.0000
0.002 0.0569 0.0569
0.004 0.1919 0.1919
0.006 0.3352 0.3354
0.008 0.4178 0.4178
0.010 0.3954 0.3954