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

20.6 Euler’s method 619
Whenx = 1,y = 1 so thatD = 1, and the analytical solution is thereforey = x. We
see that in this example the numerical solution by Euler’s method produces the exact
solution. This will always be the case when the exact solution is a linear function and
exact arithmeticisemployed.
EXERCISES20.6
1 Using Euler’s method estimatey(3) given
y ′ = x +y
x
y(2) =1
Useh = 0.5 andh = 0.25. Solvethisequation
analytically and compare your numerical solutions
with the true solution.
2 Findy(0.5)ify ′ =x +y,y(0) = 0.Useh = 0.25and
h = 0.1.Find the truesolution forcomparison.
3 Use Euler’s methodto find v(0.01)given
10 −2dv +v =sin100πt v(0) =0
dt
Takeh = 0.005 andh = 0.002. Find the analytical
solution forcomparison.
Solutions
1 Exact:y =xln |x| −0.1931x
x i y i y i y
(h = 0.5) (h = 0.25) (exact)
2.00 1.0000 1.0000 1.0000
2.25 -- 1.3750 1.3901
2.50 1.7500 1.7778 1.8080
2.75 -- 2.2056 2.2509
3.00 2.6000 2.6561 2.7165
3 Exact:
sin100πt − πcos100πt + πe−100t
v =
π 2 +1
t i v i v
(h = 0.005) (exact)
0 0.0000 0.0000
0.005 0.0000 0.2673
0.010 0.5000 0.3954
2 Exact:y=−x−1+e x
x i y i y
(h = 0.25) (exact)
0 0.0000 0.0000
0.25 0.0000 0.0340
0.50 0.0625 0.1487
x i y i y
(h = 0.1) (exact)
t i v i v
(h = 0.002) (exact)
0 0.0000 0.0000
0.002 0.0000 0.0569
0.004 0.1176 0.1919
0.006 0.2843 0.3354
0.008 0.4176 0.4178
0.010 0.4517 0.3954
0 0.0000 0.0000
0.1 0.0000 0.0052
0.2 0.0100 0.0214
0.3 0.0310 0.0499
0.4 0.0641 0.0918
0.5 0.1105 0.1487