Question

given that dy/dx = (y-x)(y+x) with initial condition y=1 for x=0 . find y for x=0.1 by Euler's method [ hints - take n = 5 i.e h = 0.02]

Answer 1

solution \space \\ given\\ find \space y(0.1)\\ for \space y' \space =(y-x)(y+x), \space y(0)=1, \space with \space step \space lenght \space 0.02 \space using \space euler \space method\\ Eulerr \space method \space \\ where \space we \space take \space y' \space as \space f(x,y)\\ [1] \space y_1(0.02) \space = \space y_0 \space +h. \space f(x_0,y_0)=1+(0.02).f(0,1)=1+(0.02).(1)=1+(0.02)=1.02\\ [2] \space y_2(0.04) \space = \space y_1 \space +h. \space f(x_1,y_1)=1.02+(0.02)f(0.02,1.02)=1.02+(0.02)⋅(1.04)=1.02+(0.0208)=1.0408\\ [3] \space y_3(0.06) \space = \space y_2 \space +h. \space f(x_2,y_2)=1.0408+(0.02)f(0.04,1.0408)=1.0408+(0.02)⋅(1.0817)=1.0408+(0.0216)=1.0624\\ [4] \space y_4(0.08) \space = \space y_3 \space +h. \space f(x_3,y_3)=1.0624+(0.02)f(0.06,1.0624)=1.0624+(0.02)⋅(1.1252)=1.0624+(0.0225)=1.0849\\ [5] \space y_5(0.1) \space = \space y_4+h. \space f(x_4,y_4)=1.0849+(0.02)f(0.08,1.0849)=1.0849+(0.02)⋅(1.1707)=1.0849+(0.0234)=1.1084\\ hence \space \\ y(0.1)=1.1084\\