

1. $$y_{1}=1.550598190, \: y_{2} = 2.469649729$$

2. $$y_{1} = 1.221551366,\: y_{2} = 1.492920208$$

3. $$y_{1} = 1.890339767,\: y_{2} = 1.763094323$$

4. $$y_{1} = 2.961316248,\: y_{2} = 2.920128958$$

5. $$y_{1} = 2.475605264,\: y_{2} = 1.825992433$$

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 $$y = ue^{−3x}$$, where $$(A) u' = 1 − 4x + 3x^{2} − 4x^{3} , u(0) = −3$$. 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.

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" $$1.50$$ $$0.343839158$$ $$0.343784814$$ $$0.343780796$$ $$0.343780513$$

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" $$3.0$$ $$16.666666988$$ $$16.666666687$$ $$16.666666668$$ $$16.666666667$$