Question

2. Find the solution of the second order differential equations: day + y = 0, y(TT/3) = 0, y'(TT/3) = dx2 a. = 4 b. y" – 8y' + 16y = 0, y(0) = 1, y(1) = 0

Answer 1

Now the complementary function is given by Solution: y' (11/3) = 4 2 (a) d²y y (7/3)=0 ty daz Now axillayny equation, mat i = 0 => m= Il ४(1) C Cosa + C sinan y' cars 2-C, sinnt la cosa = 0 71%) = a cosa + C2 sinal = » ₂24 + 3 (2=0 -V3 q C y (7/3) =-o, minst tq Cost = 4 *) - ( + C2 / 2 = 4 – 13CEZ) + ¢Ź = 4 (++) { = 4 4 4 (2= 1X2 → = 2

Get thene G = -√3 (2 =-213 Hence the solution is. YCR)= -2/5 cosn +2 sina y (2) = a sina - 285 cosa # (6 Solution :- Ciren, y"- 84' + 1by 20; y 103 = Y'(1) 20 Now Au Ausillay equation, m² 8m+16=0 mi-20400 + 4²0 >> mag=0 -4) (-4) = 0 > m=4, 4 Now the complementing fernetion Isopivan by y (2) = (6+ag eex ►y (21) = →ylla) a (que un terny + - 4Geun

Now the final general solution to (0) = c e° о.с. = 1 Ас = 1 76) — 5 90) = 4 се' + с. (че“e") = о - 4e14 ulse) = 0 ) Cse" =-4e4 - 4 С, Чи 1 3) = (1 - }}e" e4x — (s-4 %) 5