Question

cos 4t 1:[0,) 3. Solve for y(t): y” +16y = f(t) = { with y(0) = 0 and y'(0) = 0. 0,if tn. Saleserstos se va posar este mai 90 -m70-e Stepl. Answer: y(t) =

Answer 1

Given differential equation is y4164 Joht, t:[0,1). with the initial conditions yo) 20 lo, it to F yllo)=0 To find the Complementary Function . Consider the corresponding homogenous differential aquation ytky co with the conditions 410320, y110) 20 Let y = cena be trial solration. any"=cme mu Implementing these in the differential equation we get cmtemu 1660mxo » cemx (m + 16) 20 m71620 > m= t4i y = q Cosat + q sinat Now, yo)=0 7420 y/10)20 7 ₂ 4 Cosatto » C=0 - complementarly solution is yro

To find particular solution Let D = 1 D² = 1 Hence the particulare solution is i cosat ,if teto, H & D 16 (0) it tytt Noco, bio COSA - Ceritetit, Now, the epit fit - Rule: Ice Cemku 7 a e (D+11) 16 = e me & CD M (V) 1416 si by Di ko = edit (1+ ' 2 = alt Ś CI-D) x si similarly, I 16 e 7 x _ - 2 unit Cosat = teretit te ting a 4 - 3 2 1 1 Cenit ait) - 2 sinat

Also, Dalis (0) Hence the particular solution is ylt) = 8 h - 2 singt, if tE to, ) o sit tytt Finally, the solution of the given equation is - complementary solution - Particulars solution - ot & t singt, THE 10,41) lo, it ty of = 2 + sinkt, it teto, t) けもみ