Question

dy: 2 Consider the following Ordinary Differential Equation (ODE) for function yı(z) on interval [0, 1] +(-10,3) dayi dy + 28.06 + (-16.368) + y(x) = 1.272.0.52 with the following initial conditions at point a = 0; dy 91 = 4.572 = 30.6248 = 185.2223 dar Introducting notations dyi dy2 dy dar dar dir? convert the ODE to the system of three first-order ODEs for functions y1, y2, y3 in the form: dy dar fi (1, y1, ya, y) dy2 de 12(2, 41, 42, 43) dys =f(x, y, y2, 33) dar Apply Euler's method with h = 0.5 to solve this system numerically. Fill in the blank spaces in the following table. Round up your answers to 4 decimals. yi y2 O ys 10.5 1.0

Answer 1

doy dx 4100 = 4.572 (10.3) dry 0.92 dx +(28.06) de - 16:368 y= 1.879 e L O I (0) = 30.6248 dexam) = 185.2223 dyi Consider Yg - The dy dya - de 41 dya dy Y a dya 43 and 0 = che dY₃ dx 09x 1.9726 + 16.3684,- (28.06) Ya so we get + ((0.3) 43 dy dya dx dy dx 93 -f08) 1.272€ + 16:3684-98.06427 0.9x. 4:00) = 4.572 4260) - di () = 30-6248 10.343 dx y300- do dy dx (0) = 185.2223 get we X(0) = x = f(x) 4.572 30.6248 h=0.5 Euler's Method: L 185.2223 X (th) = X6.)+ f(x) At X=0; =0.53) X(0,5) = - u. 572 30.6248 130.6248 + (0.5185.2223 185.223 1124.564898 X(0.5) - y (0.5) 19.8844 19.88447 123.93595 -) - 797. S044ug Yg (0.5) = 123.23595 4360.5) - 707. Solug

At x=05 ; R=0.5 XI - X(0.5) + (0-5) f(41(0-5), (0.5), 40(05)) X(0) = [ 19.8844 123.23595 123.23595 + (0.5) 747. souuug 747. souuug 4568.75782 - X (1) = 181.508375 ug 6.9881745 3031883359 y (1) = 81. S09375 Y W = 496.9881845 4360 = 3631-883359 Table X, 43 185.2223 0 14.572 30.6248 0.5 19.8844 123 - 23595 1747.564429 81.508375 496.9881745 3031.883359 Answer