

1. $$y_{1}= 1.450000000,\: y_{2} = 2.085625000,\: y_{3} = 3.079099746$$

2. $$y_{1} = 1.200000000,\: y_{2} = 1.440415946,\: y_{3} = 1.729880994$$

3. $$y_{1} = 1.900000000,\: y_{2} = 1.781375000,\: y_{3} = 1.646612970$$

4. $$y_{1} = 2.962500000,\: y_{2} = 2.922635828,\: y_{3} = 2.880205639$$

5. $$y_{1} = 2.513274123,\: y_{2} = 1.814517822,\: y_{3} = 1.216364496$$

6.

 $$x$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ Exact $$1.0$$ $$48.298147362$$ $$51.492825643$$ $$53.076673685$$ $$54.647937102$$

7.

 $$x$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ Exact $$2.0$$ $$1.390242009$$ $$1.370996758$$ $$1.361921132$$ $$1.353193719$$

8.

 $$x$$ $$h=0.05$$ $$h=0.025$$ $$h=0.0125$$ Exact $$1.50$$ $$7.886170437$$ $$8.852463793$$ $$9.548039907$$ $$10.500000000$$

9.

 $$x$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ $$3.0$$ $$1.469458241$$ $$1.462514486$$ $$1.459217010$$ $$0.3210$$ $$0.1537$$ $$0.0753$$ Approximate Solutions Residuals

10.

 $$x$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ $$2.0$$ $$0.473456737$$ $$0.483227470$$ $$0.487986391$$ $$-0.3129$$ $$-0.1563$$ $$-0.0781$$ Approximate Solutions Residuals

11.

 $$x$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ "Exact" $$1.0$$ $$0.691066797$$ $$0.676269516$$ $$0.668327471$$ $$0.659957689$$

12.

 $$x$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ "Exact" $$2.0$$ $$-0.772381768$$ $$-0.761510960$$ $$-0.756179726$$ $$-0.750912371$$

13.

 Euler's Method $$x$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ Exact $$1.0$$ $$0.538871178$$ $$0.593002325$$ $$0.620131525$$ $$0.647231889$$

 Euler semilinear Method $$x$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ Exact $$1.0$$ $$0.647231889$$ $$0.647231889$$ $$0.647231889$$ $$0.647231889$$

Applying variation of parameters to the given initial value problem yields $$y = ue^{−3x}$$ , where (A) $$u' = 7, u(0) = 6$$. Since $$u''= 0$$, Euler’s method yields the exact solution of (A). Therefore the Euler semilinear method produces the exact solution of the given problem

14.

 Euler's Method $$x$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ "Exact" $$3.0$$ $$12.804226135$$ $$13.912944662$$ $$14.559623055$$ $$15.282004826$$

 Euler semilinear method $$x$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ "Exact" $$3.0$$ $$15.354122287$$ $$15.317257705$$ $$15.299429421$$ $$15.282004826$$

15.

 Euler's method $$x$$ $$h=0.2$$ $$h=0.1$$ $$h=0.05$$ "Exact" $$2.0$$ $$0.867565004$$ $$0.885719263$$ $$0.895024772$$ $$0.904276722$$

 Euler's semilinear method $$x$$ $$h=0.2$$ $$h=0.1$$ $$h=0.05$$ "Exact" $$2.0$$ $$0.569670789$$ $$0.720861858$$ $$0.808438261$$ $$0.904276722$$

16.

 Euler's method $$x$$ $$h=0.2$$ $$h=0.1$$ $$h=0.05$$ "Exact" $$3.0$$ $$0.922094379$$ $$0.945604800$$ $$0.956752868$$ $$0.967523153$$

 Euler semilinear method $$x$$ $$h=0.2$$ $$h=0.1$$ $$h=0.05$$ "Exact" $$3.0$$ $$0.993954754$$ $$0.980751307$$ $$0.974140320$$ $$0.967523153$$

17.

 Euler's method $$x$$ $$h=0.0500$$ $$h=0.0250$$ $$h=0.0125$$ "Exact" $$1.50$$ $$0.319892131$$ $$0.330797109$$ $$0.337020123$$ $$0.343780513$$

 Euler semilinear method $$x$$ $$h=0.0500$$ $$h=0.0250$$ $$h=0.0125$$ "Exact" $$1.50$$ $$0.305596953$$ $$0.323340268$$ $$0.333204519$$ $$0.343780513$$

18.

 Euler's method $$x$$ $$h=0.2$$ $$h=0.1$$ $$h=0.05$$ "Exact" $$2.0$$ $$0.754572560$$ $$0.743869878$$ $$0.738303914$$ $$0.732638628$$

 Euler semilinear method $$x$$ $$h=0.2$$ $$h=0.1$$ $$h=0.05$$ "Exact" $$2.0$$ $$0.722610454$$ $$0.727742966$$ $$0.730220211$$ $$0.732638628$$

19.

 Euler's method $$x$$ $$h=0.0500$$ $$h=0.0250$$ $$h=0.0125$$ "Exact" $$1.50$$ $$2.175959970$$ $$2.210259554$$ $$2.227207500$$ $$2.244023982$$

 Euler semilinear method $$x$$ $$h=0.0500$$ $$h=0.0250$$ $$h=0.0125$$ "Exact" $$1.50$$ $$2.117953342$$ $$2.179844585$$ $$2.211647904$$ $$2.244023982$$

20.

 Euler's method $$x$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ "Exact" $$1.0$$ $$0.032105117$$ $$0.043997045$$ $$0.050159310$$ $$0.056415515$$

 Euler's semilinear method $$x$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ "Exact" $$1.0$$ $$0.056020154$$ $$0.056243980$$ $$0.056336491$$ $$0.056415515$$

21.

 Euler's method $$x$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ "Exact" $$1.0$$ $$28.987816656$$ $$38.426957516$$ $$45.367269688$$ $$54.729594761$$

 Euler's semilinar method $$x$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ "Exact" $$1.0$$ $$54.709134946$$ $$54.724150485$$ $$54.728228015$$ $$54.729594761$$

22.

 Euler's method $$x$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ "Exact" $$3.0$$ $$1.361427907$$ $$1.361320824$$ $$1.361332589$$ $$1.361383810$$

 Euler's semilinar method $$x$$ $$h=0.1$$ $$h=0.05$$ $$h=0.025$$ "Exact" $$3.0$$ $$1.291345518$$ $$1.326535737$$ $$1.344004102$$ $$1.361383810$$