Question

6.3.2. A. Write the given IVP as a system. Then do two steps of Euler's method by hand (perhaps with a calculator) with the indicated step size h. Using the given exact solution, compute the error after the second step. (d) 2x2У" + 3xy'-y=0,y(1)= 4, y(1)=-1.)(x)=2(x1/2+x-1),b=1/4

Answer 1

2x2 y' + 3 χ y'--0 with boundary conditions Let y,-y and y,-y'. Ћеп 2 2. 2. 2x2 2 x Thus, the System of 2 First order ode's is 2 y2 (1)--1 4, yl (1) such that, The Euleγ method is given as lh) For n: O 4 SO Here,

and, F(x, g) - 2z2 SO, 14 31 (-1) 7/2 f ( ỵ,' y(o)) In egn. (1), to obtain and Substitute y 4-1 4- 4 412 15/4 , this implies, y."- 5 and y,"- 4 Now, for n- 1 4 4-

y2 y1( 1)一321;" 2χ 1/ 9 4.2 { 15/4-3x(5/4)X (-1/8)} 2 (5/4)2 21/20 -118 15/4 So, y2) 4 21/20 119/32 15/4 - 1/3 2 1/80 /82/8 , this implies, y (2)-119 32 119/32 1/80 2) 42 80 since, t, -y, therepore, the e stimate to y' after 2 iterat 12)-119 = 3.713 ons IS 32 and the actual solution at y=4 is given by the equation

JS +B-= 3.836 so, the error after 2nd iteration is given as g | = C2) | 3 구19-3836 4 こ1.17x10-1