Question

S ? 2 for even intervals of width = 1 (0,1), (2,3), (3,4) .... 1 for odd intervals of width = 1 [1,2], [3,4], [5,6] ....

Answer 1

Mass - Spring /ODE ..or even internal f(t) = {2 , t€ (0, 1) or (2,3) or (4, 5) or, Il tt [1,2]o (3,47 or [5,6) or, .... odd interval 2 3 4 5 ODE x +5x = f(t) - can be splied into two region where f(t) = 1 and 1 (ie evenodd 2 so a" + 5x=2- " + 5%-14 t t even interval tt Odd interval Let a= ent be a trial sold or homogenious parl- So, m2 + 5 = 0 -> n tirs so, Ye = G cosrot + ₂ sinust @ G, e are this is homogenious ssolr arbitrary for both constant of the ① and ② for Now we find Up for ③ Yp = 245 = 25 Similarly particular soln for 1 is Yp=Y5 .

Input function f(t)

homogeneous soln in even interval (tepeted) - 02 0.4 0.6 0.8 i

homogenious part of the solution (both odd and even interval)

particular solution

:. The solutions are in even interval y = acos 55t +₂ Sindst + 2 / - (3) in odd 7 = e, cos 55 t + C sin 55t + V (32) Initial condition 70)=1 (161) y'(0)=1 -IC2) From from (1) and (3:1) » ci + 2 = 1 > a = 3 Ic2 and (31) => C2 √5 + 0 = 1 cz = 'te 5 456 from (TC1) and (3.2) - e, at 1/3 =1 from rc2 and (3.2 > eqr5 =1 > G = 46 l = 1/2 Finally the solutions Even interval are in - y = 3/5 cos15t + √5 Sin 55. Odd interval y = 4/3 cos15t + Is Sinot th e