Question

two-step explicit and implicit me ollowing differential equation with y1) 1,6487 and yt1 05)-1 7354 USE FOUR DECIMAL DIG Use t thods to apprcximate y(1.1) for the f with y(1) 1 6487 and y(1.05) 1.7354 USE FOUR DEOIMAL DIGIT ROUNDING dy dy

Answer 1

Given dy/dx = xy y^1 = xy with y (1) = 1.6487 y (1.05) = 1.7354 let f (x, y) , x_0 = 1 y_0 = 1.6487 h = 0.025 By fourth order Runge kutta method k_1 = h f (x_0, y_0) = f x_0 y_0 = 0.0412 k_2 = h f (x_0 + h/2, y_0 + k_1/2) = h (x_0 + h/2) (y_0 + k_2/2) = 0.0423 k_4 = h f (x_0 + h, y_0 + k_3) = h (x_0 + h) (y_0 + k_3) = 0.0433 y_1 = y (1.025) = y_0 + 1/6 (k_0 + 2k_2 + 2k_3 + k_4) = 1.6910 Given that y_2 = y (1.05) = 1.7354 implies x_2 = 1.05 again by fourth order Runge kutta method k_1 = hf (x_2, y_2) = hx_2 y_2 = 0.0456 k_2 = hf (x_2 + h/2, y_2 + k_2/2) = h (x_2 + h/2) (y_2 + k_1/2) = 0.046 k_3 = hf (x_2 + h/2, y_2 + k_3) = 0.0467 k_4 = hf (x_2 + h, y_2 + k_3 = 0.0479 y_3 = y (1.075) = y_2 + 1/6 (k_1 + 2k_2 + 2k_3 + k_4) = 1. .