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A.3.3: Section 3.3 Answers

( \newcommand{\kernel}{\mathrm{null}\,}\)

1. y1=1.550598190,y2=2.469649729

2. y1=1.221551366,y2=1.492920208

3. y1=1.890339767,y2=1.763094323

4. y1=2.961316248,y2=2.920128958

5. y1=2.475605264,y2=1.825992433

6.

x h=0.1 h=0.05 h=0.025 Exact
1.0 54.654509699 54.648344019 54.647962328 54.647937102

7.

x h=0.1 h=0.05 h=0.025 Exact
2.0 1.353191745 1.353193606 1.353193712 1.353193719

8.

x h=0.05 h=0.025 h=0.0125 Exact
1.50 10.498658198 10.499906266 10.499993820 10.500000000

9.

x h=0.1 h=0.05 h=0.025 h=0.1 h=0.05 h=0.025
3.0 1.456023907 1.456023403 1.456023379 0.0000124 0.000000611 0.0000000333
  Approximate Solutions Residuals

10.

x h=0.1 h=0.05 h=0.025 h=0.1 h=0.05 h=0.025
2.0 0.492663789 0.492663738 0.492663736 0.000000902 0.0000000508 0.00000000302
  Approximate Solutions Residuals

11.

x h=0.1 h=0.05 h=0.025 "Exact"
1.0 0.659957046 0.659957646 0.659957686 0.659957689

12.

x h=0.1 h=0.05 h=0.025 "Exact"
2.0 0.750911103 0.750912294 0.750912367 0.750912371

13. Applying variation of parameters to the given initial value problem yields y=ue3x, where (A)u=14x+3x24x3,u(0)=3. Since u(5)=0, the Runge-Kutta method yields the exact solution of (A). Therefore the Euler semilinear method produces the exact solution of the given problem.

clipboard_e9e428930ffce428ad2d7df8d39937a25.png

clipboard_e075f364cb890bde142bc6294dc1ba24e.png

14.

Runge-Kutta method
x h=0.1 h=0.05 h=0.025 "Exact"
3.0 15.281660036 15.281981407 15.282003300 15.282004826
Runge-Kutta semilinear method
x h=0.1 h=0.05 h=0.025 "Exact"
3.0 15.282005990 15.282004899 15.282004831 15.282004826

15.

Runge-Kutta method
x h=0.2 h=0.1 h=0.05 "Exact"
2.0 0.904678156 0.904295772 0.904277759 0.904276722
Runge-Kutta semilinear method
x h=0.2 h=0.1 h=0.05 "Exact"
2.0 0.904592215 0.904297062 0.904278004 0.904276722

16.

Runge-Kutta method
x h=0.2 h=0.1 h=0.05 "Exact"
3.0 0.967523147 0.967523152 0.967523153 0.967523153
Runge-Kutta semilinear method
x h=0.2 h=0.1 h=0.05 "Exact"
3.0 0.967523147 0.967523152 0.967523153 0.967523153

17.

Runge-Kutta method
x h=0.0500 h=0.0250 h=0.0125 "Exact"
1.50 0.343839158 0.343784814 0.343780796 0.343780513

clipboard_e7cd069c28f0e49fea809ca150442f812.png

18.

Runge-Kutta method
x h=0.2 h=0.1 h=0.05 "Exact"
2.0 0.732633229 0.732638318 0.732638609 0.732638628
Runge-Kutta semilinear method
x h=0.2 h=0.1 h=0.05 "Exact"
2.0 0.732639212 0.732638663 0.732638630 0.732638628

19.

Runge-Kutta method
x h=0.0500 h=0.0250 h=0.0125 "Exact"
1.50 2.244025683 2.244024088 2.244023989 2.244023982
Runge-Kutta semilinear method
x h=0.0500 h=0.0250 h=0.0125 "Exact"
1.50 2.244025081 2.244024051 2.244023987 2.244023982

20.

Runge-Kutta method
x h=0.1 h=0.05 h=0.025 "Exact"
1.0 0.056426886 0.056416137 0.056415552 0.056415515
Runge-Kutta semilinear method
x h=0.1 h=0.05 h=0.025 "Exact"
1.0 0.056415185 0.056415495 0.056415514 0.056415515

21.

Runge-Kutta method
x h=0.1 h=0.05 h=0.025 "Exact"
1.0 54.695901186 54.727111858 54.729426250 54.729594761
Runge-Kutta semilinear method
x h=0.1 h=0.05 h=0.025 "Exact"
1.0 54.729099966 54.729561720 54.729592658 54.729594761

22.

Runge-Kutta method
x h=0.1 h=0.05 h=0.025 "Exact"
3.0 1.361384082 1.361383812 1.361383809 1.361383810
Runge-Kutta semilinear method
x h=0.1 h=0.05 h=0.025 "Exact"
3.0 1.361456502 1.361388196 1.361384079 1.361383810

24.

x h=.1 h=.05 h=.025 Exact
2.00 1.000000000 1.000000000 1.000000000 1.000000000

25.

x h=.1 h=.05 h=.025 "Exact"
1.00 1.000000000 1.000000000 1.000000000 1.000000000

26.

x h=.1 h=.05 h=.025 "Exact"
1.50 4.142171279 4.142170553 4.142170508 4.142170505

27.

x h=.1 h=.05 h=.025 "Exact"
3.0 16.666666988 16.666666687 16.666666668 16.666666667

This page titled A.3.3: Section 3.3 Answers is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by William F. Trench.

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