Question

Consider the following initial value problem у(0) — 0. у%3D х+ у, (i) Solve the differential equation above in tabular form with h= 0.2 to approximate the solution at x=1 by using Euler's method. Give your answer accurate to 4 decimal places. Given the exact solution of the differential equation above is y= e-x-1. Calculate (ii) all the error and percentage of relative error between the exact and the approximate y values for each of values in (i) 0.2 0.4 0.6 0.8 1.0 X 0.2221 0.4255 0.0214 0.0918 0.7183 y Table 1 Table 1 shows approximation solution for differential equation above by using 4th order Runge-Kutta method. (iii Based on the table above, sketch a graph showing the lines of exact and both approximate y values. (iv) Based on your observation on the graph in (iii), which method is more Give an appropriate reason. accurate?

Answer 1

h=0.2 usng Eulor mathad y' t - SohxoJ 0+02(0+0) O.2+0 =0.2 X1= 0.2, Y,=D (261) 4 IR = 0.2 (0.2) = 0.04 y2 = 440-04 =0.44 X2=0.4, 20.04 .04 4 0.088 =Q.122 93 = D-04 + 0.2 (0.44) 3 .6, Ya= 0-3 C3,a)= 0.6+0.128 0.T28 6.128+0.2 (0.792)= 0.2736 La y=.8,u= 0.2736 Cu,) 0.8+0.2736 1.0736 gutht (u o2736+0.2 (1.0736) S0.2736+0.2147 =.4883 E Res mathedExad ReaTiwe error Error I フ) Y O0214 10D= 100 0.0214 0OR14 02 0.0513 100= 56.43%. 0919 9.094l.100 42.37 2221 OISI9.1oO 0.0S1% 0.0918 04 0.04 0.222 0.0941 0.128 35.7 0.8 0.2736 0.42SS 0-1S19 O.23 32/ 1.0 0.4883 0.783 0.2300 ti8l00= (ii) Baed onthe table gran skatchof nos a exact and looth apbroximata y valluon an graph shach Civ Tho mont accurate mathed is thordor Runge kutfe Method Banca ites tha very clone ndution and Jraph a exat nolutrah and Runge Kuta nolutial u idartical

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