$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

1. $$y_{1} = 1.542812500,\: y_{2} = 2.421622101,\: y_{3} = 4.208020541$$

2. $$y_{1} = 1.220207973,\: y_{2} = 1.489578775.\: y_{3} = 1.819337186$$

3. $$y_{1} = 1.890687500,\: y_{2} = 1.763784003,\: y_{3} = 1.622698378$$

4. $$y_{1} = 2.961317914,\: y_{2} = 2.920132727,\: y_{3} = 2.876213748$$

5. $$y_{1} = 2.478055238,\: y_{2} = 1.844042564,\: y_{3} = 1.313882333$$

6.

 $$x$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ Exact $$1.0$$ $$56.134480009$$ $$55.003390448$$ $$54.734674836$$ $$54.647937102$$

7.

 $$x$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ Exact $$2.0$$ $$1.353501839$$ $$1.353288493$$ $$1.353219485$$ $$1.353193719$$

8.

 $$x$$ $$h=0.5$$ $$h=0.025$$ $$h=0.0125$$ Exact $$1.50$$ $$10.141969585$$ $$10.396770409$$ $$10.472502111$$ $$10.500000000$$

9.

 $$x$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ $$3.0$$ $$1.455674816$$ $$1.455935127$$ $$1.456001289$$ $$-0.00818$$ $$-0.00207$$ $$-0.000518$$ Approximate Solutions Residuals

10.

 $$x$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ $$2.0$$ $$0.492862999$$ $$0.492709931$$ $$0.492674855$$ $$0.00335$$ $$0.000777$$ $$0.000187$$ Approximate Solutions Residuals

11.

 $$x$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ "Exact" $$1.0$$ $$0.660268159$$ $$0.660028505$$ $$0.659974464$$ $$0.659957689$$

12.

 $$x$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ "Exact" $$2.0$$ $$-0.749751364$$ $$-0.750637632$$ $$-0.750845571$$ $$-0.750912371$$

13. Applying variation of parameters to the given initial value problem $$y = ue^{−3x}$$ , where $$(A) u' = 1 − 2x, u(0) = 2$$. Since $$u''' = 0$$, the improved Euler method yields the exact solution of (A). Therefore the improved Euler semilinear method produces the exact solution of the given problem.

 Improved Euler method $$x$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ Exact $$1.0$$ $$0.105660401$$ $$0.100924399$$ $$0.099893685$$ $$0.099574137$$

 Improved Euler semilinar method $$x$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ Exact $$1.0$$ $$0.099574137$$ $$0.099574137$$ $$0.099574137$$ $$0.099574137$$

14.

 Improved Euler method $$x$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ "Exact" $$3.0$$ $$15.107600968$$ $$15.234856000$$ $$15.269755072$$ $$15.282004826$$

 Improved Euler semilinar method $$x$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ "Exact" $$3.0$$ $$15.285231726$$ $$15.282812424$$ $$15.282206780$$ $$15.282004826$$

15.

 Improved Euler method $$x$$ $$h=0.2$$ $$h=0.1$$ $$h=0.05$$ "Exact" $$2.0$$ $$0.924335375$$ $$0.907866081$$ $$0.905058201$$ $$0.904276722$$

 Improved Euler semilinear method $$x$$ $$h=0.2$$ $$h=0.1$$ $$h=0.05$$ "Exact" $$2.0$$ $$0.969670789$$ $$0.920861858$$ $$0.908438261$$ $$0.904276722$$

16.

 Improved Euler method $$x$$ $$h=0.2$$ $$h=0.1$$ $$h=0.05$$ "Exact" $$3.0$$ $$0.967473721$$ $$0.967510790$$ $$0.967520062$$ $$0.967523153$$

 Improved Euler semilinear method $$x$$ $$h=0.2$$ $$h=0.1$$ $$h=0.05$$ "Exact" $$3.0$$ $$0.967473721$$ $$0.967510790$$ $$0.967520062$$ $$0.967523153$$

17.

 Improved Euler method $$x$$ $$h=0.0500$$ $$h=0.0250$$ $$h=0.0125$$ "Exact" $$1.50$$ $$0.349176060$$ $$0.345171664$$ $$0.344131282$$ $$0.343780513$$

 Improved Euler semilinear method $$x$$ $$h=0.0500$$ $$h=0.0250$$ $$h=0.0125$$ "Exact" $$1.50$$ $$0.349350206$$ $$0.345216894$$ $$0.344142832$$ $$0.343780513$$

18.

 Improved Euler method $$x$$ $$h=0.2$$ $$h=0.1$$ $$h=0.05$$ "Exact" $$2.0$$ $$0.732679223$$ $$0.732721613$$ $$0.732667905$$ $$0.732638628$$

 Improved Euler semilinear method $$x$$ $$h=0.2$$ $$h=0.1$$ $$h=0.05$$ "Exact" $$2.0$$ $$0.732166678$$ $$0.732521078$$ $$0.732609267$$ $$0.732638628$$

19.

 Improved Euler method $$x$$ $$h=0.0500$$ $$h=0.0250$$ $$h=0.0125$$ "Exact" $$1.50$$ $$2.247880315$$ $$2.244975181$$ $$2.244260143$$ $$2.244023982$$

 Improved Euler semilinear method $$x$$ $$h=0.0500$$ $$h=0.0250$$ $$h=0.0125$$ "Exact" $$1.50$$ $$2.248603585$$ $$2.245169707$$ $$2.244310465$$ $$2.244023982$$

20.

 Improved Euler method $$x$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ "Exact" $$1.0$$ $$0.059071894$$ $$0.056999028$$ $$0.056553023$$ $$0.056415515$$

 Improved Euler semilinear method $$x$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ "Exact" $$1.0$$ $$0.056295914$$ $$0.056385765$$ $$0.056408124$$ $$0.056415515$$

21.

 Improved Euler method $$x$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ "Exact" $$1.0$$ $$50.534556346$$ $$53.483947013$$ $$54.391544440$$ $$54.729594761$$

 Improved Euler semilinear method $$x$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ "Exact" $$1.0$$ $$54.709041434$$ $$54.724083572$$ $$54.728191366$$ $$54.729594761$$

22.

 Improved Euler method $$x$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ "Exact" $$3.0$$ $$1.361395309$$ $$1.361379259$$ $$1.361382239$$ $$1.361383810$$

 Improved Euler semilinear method $$x$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ "Exact" $$3.0$$ $$1.375699933$$ $$1.364730937$$ $$1.362193997$$ $$1.361383810$$

23.

 $$x$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ Exact $$2.0$$ $$1.349489056$$ $$1.352345900$$ $$1.352990822$$ $$1.353193719$$

24.

 $$x$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ Exact $$2.0$$ $$1.350890736$$ $$1.352667599$$ $$1.353067951$$ $$1.353193719$$

25.

 $$x$$ $$h=0.05$$ $$h=0.025$$ $$h=0.0125$$ Exact $$1.50$$ $$10.133021311$$ $$10.391655098$$ $$10.470731411$$ $$10.500000000$$

26.

 $$x$$ $$h=0.05$$ $$h=0.025$$ $$h=0.0125$$ Exact $$1.50$$ $$10.136329642$$ $$10.393419681$$ $$10.470731411$$ $$10.500000000$$

27.

 $$x$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ "Exact" $$1.0$$ $$0.660846835$$ $$0.660189749$$ $$0.660016904$$ $$0.659957689$$

28.

 $$x$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ "Exact" $$1.0$$ $$0.660658411$$ $$0.660136630$$ $$0.660002840$$ $$0.659957689$$

29.

 $$x$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ "Exact" $$2.0$$ $$-0.750626284$$ $$-0.750844513$$ $$-0.750895864$$ $$-0.751331499$$

30.

 $$x$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ "Exact" $$2.0$$ $$-0.750335016$$ $$-0.750775571$$ $$-0.750879100$$ $$-0.751331499$$