1.3 Example:

Use the Euler's Algorithm to find the approximate value of the solution of the IVP

$$y'=y^2,\qquad y(0)=1,\qquad 0\leq x \leq 0.5,$$

using step size (i)0.1, (ii)0.005, (iii) (iv)
Show that the exact solution of the IVP is $y(x)=\frac{1}{1-x}.$ Calculate the error and tabulate the results.
Solution: Comparing the given IVP with (1.1), we note that $f(x,y)=y^2, \,\, a=0,\,b=0.5,\, y_0=1$. The Euler's algorithm now reads ..... used notations,

$$y_{k+1}=y_k+hy^2_k,\qquad k=0,1,2...$$and $$y_0=1$$

See tables 1,2,3 and 4 for the results (upto 5 decimal places)

		Table 1
Initial x	Initial y	Stepsize h	Appx. y	Exa. y	Error
0.00000	1.00000	0.10000
0.10000	1.00000	0.10000	1.10000	1.11111	0.01111
0.20000	1.10000	0.10000	1.22100	1.25000	0.02900
0.30000	1.22100	0.10000	1.37008	1.42857	0.05849
0.40000	1.37008	0.10000	1.55780	1.66667	0.10887
0.50000	1.55780	0.10000	1.80047	2.00000	0.19953

		Table 2
Initial x	Initial y	Stepsize h	Appx. y	Exa. y	Error
0.00000	1.00000	0.05000
0.05000	1.00000	0.05000	1.05000	1.05263	0.00263
0.10000	1.05000	0.05000	1.10513	1.11111	0.00599
0.15000	1.10513	0.05000	1.16619	1.17647	0.01028
0.20000	1.16619	0.05000	1.23419	1.25000	0.01581
0.25000	1.23419	0.05000	1.31035	1.33333	0.02298
0.30000	1.31035	0.05000	1.39620	1.42857	0.03237
0.35000	1.39620	0.05000	1.49367	1.53846	0.04479
0.40000	1.49367	0.05000	1.60522	1.66667	0.06144
0.45000	1.60522	0.05000	1.73406	1.81818	0.08412
0.50000	1.73406	0.05000	1.88441	2.00000	0.11559

		Table 3
Initial x	Initial y	Stepsize h	Appx. y	Exa. y	Error
0.00000	1.00000	0.02500
0.02500	1.00000	0.02500	1.02500	1.02564	0.00064
0.05000	1.02500	0.02500	1.05127	1.05263	0.00137
0.07500	1.05127	0.02500	1.07889	1.08108	0.00219
0.10000	1.07889	0.02500	1.10799	1.11111	0.00312
0.12500	1.10799	0.02500	1.13869	1.14286	0.00417
0.15000	1.13869	0.02500	1.17110	1.17647	0.00537
0.17500	1.17110	0.02500	1.20539	1.21212	0.00673
0.20000	1.20539	0.02500	1.24171	1.25000	0.00829
0.22500	1.24171	0.02500	1.28026	1.29032	0.01006
0.25000	1.28026	0.02500	1.32124	1.33333	0.01210
0.27500	1.32124	0.02500	1.36488	1.37931	0.01443
0.30000	1.36488	0.02500	1.41145	1.42857	0.01712
0.32500	1.41145	0.02500	1.46125	1.48148	0.02023
0.35000	1.46125	0.02500	1.51464	1.53846	0.02383
0.37500	1.51464	0.02500	1.57199	1.60000	0.02801
0.40000	1.57199	0.02500	1.63377	1.66667	0.03290
0.42500	1.63377	0.02500	1.70050	1.73913	0.03863
0.45000	1.70050	0.02500	1.77279	1.81818	0.04539
0.47500	1.77279	0.02500	1.85136	1.90476	0.05340
0.50000	1.85136	0.02500	1.93705	2.00000	0.06295

		Table 4
Initial x	Initial y	Stepsize h	Appx. y	Exa. y	Error
0	1	0.0250000
0.0250000	1	0.0250000	1.025	1.025641026	0.000641026
0.0500000	1.025	0.0250000	1.051265625	1.052631579	0.001365954
0.0750000	1.051265625	0.0250000	1.07889461	1.081081081	0.002186471
0.1000000	1.07889461	0.0250000	1.10799495	1.111111111	0.003116161
0.1250000	1.10799495	0.0250000	1.13868627	1.142857143	0.004170873
0.1500000	1.13868627	0.0250000	1.171101431	1.176470588	0.005369158
0.1750000	1.171101431	0.0250000	1.205388395	1.212121212	0.006732817
0.2000000	1.205388395	0.0250000	1.241712424	1.25	0.008287576
0.2250000	1.241712424	0.0250000	1.280258668	1.290322581	0.010063913
0.2500000	1.280258668	0.0250000	1.321235224	1.333333333	0.012098109
0.2750000	1.321235224	0.0250000	1.364876787	1.379310345	0.014433558
0.3000000	1.364876787	0.0250000	1.411449003	1.428571429	0.017122425
0.3250000	1.411449003	0.0250000	1.46125371	1.481481481	0.020227771
0.3500000	1.46125371	0.0250000	1.514635271	1.538461538	0.023826268
0.3750000	1.514635271	0.0250000	1.571988271	1.6	0.028011729
0.4000000	1.571988271	0.0250000	1.633766949	1.666666667	0.032899718
0.4250000	1.633766949	0.0250000	1.70049681	1.739130435	0.038633625
0.4500000	1.70049681	0.0250000	1.772789045	1.818181818	0.045392773
0.4750000	1.772789045	0.0250000	1.85135857	1.904761905	0.053403335
0.5000000	1.85135857	0.0250000	1.937046784	2	0.062953216

We conclude this section with a flow chart for Euler's Algorithm.