Question

ordinary D ffeen tal tquations oDe Pact a) 0so Eler's Method to sohe tne oE albore from 1.0 to 2.o wth step st2e of o.s.The ntial condltrono) 2 what isyour e Hoon's Method to salve the cquatian above from Lo to 2.o using a step size of o.s. The nitial condi tion is ylio) terate on the corrector once (or each step. What is your eshnrate ot the y vaue ato Uhe itve Clossial Four th Order Ronge kot Metud to solve the ODE above fromm o .ltoー2.0-with astep sıze at2. The initial condtion is ulo)what us your ashimate of the y vmne atx 2.o?

class: Numerical Methods
solve question 1 entirely includes parts a,b and c.
the correct answers are
a) 1.545
b)1.602
c)2.031

Answer 1

dy/dx = cos y * x^2 = f(x, y) Using Euler's method First approximation y_1 = y_0 + hf (x_0, y_0) [using formula y_n + 1 = y-n + h (x_n + 1) = 2 + 0.5 f(1, 2) = 2 + 0.5 times (-0.416) [Here cos 2 = -0.416 [radius] = 1.792 Second approximation y_2 = y_1 + 0.5 f(x_1, y_1) = 1.792 + 0.5 f(1.5, 1.792) = 1.792 + 0.5 times (-0.219) times 2.25 = 1.792 - 0.2463 = 1.5457 \r\n dy/dx = cos y * x^2 = f(x, y) Using Method y^p_1 = y_0 + h f(x_0, y_0) = 1 + 0.5 f(1, 1) = 1 + 0.5 times cos 1 times 1 [cos 1 = 0.5403 in radian] = 1 + 0.5 times 0.5403 = 1 + 0.2701 = 1.2701 y^c_1 = y_0 + h/2 {f(x_0, y_0) + f(x-1, y, p)} = 1 + 0.5/2 {0.5403 + f(1.9, 1.2701)} = 1 + 0.5/2 {0.5403 + (2961) times (2.25)}