Question

(10) Solve y" + 6y' + 25 y=te-2t with y(0) = 3 and and y'(0) = 8. Describe the long term behavior of the system with some discussion of the time scale.

Answer 1

Cliven that y"+ 69 +284 = t ezt y (0) = 3, y lo] =8 for Homogeneous paut Y"+6914254 so the Aumlary equation is leiven By: m²t6m + 2520 2) m = -6 + J 36-4x25 2 3-6. 1 36-100 =-6281 2 5678 me 1673 -6700 re m = -3+41 m2 = - 3-ui purly Complea rent & the complemetry soln it liven my yees er et [ci eus atp Cg sin4t] For particular Integral Define f(0) = 8 +60 7.25 t e 2t P. I n netop+25

iss we have formula peut seperti tem" wa) = con ven f(0+2) .eve | د ا . --2 D7604211 V1A= as (0-252 +6 (0-2) + 25 :50+4-4D +6D-12+25 ED?+2D +17 I 8+ = 24 t (D?+2D+17) D²760725 po f t - *«2? = 1-8) - 28* 7. 172 so suin is yest) + p •[: 1764) = 2 * (41 tus ger Cabin 4 t) + As y co)3 2) 21- 23 3 + 2 ei I 172

WWW. IM H OWUWWWWWWWWWWWWWWWWWWWWWWWWWWWW WW!! 3x289 f + y coj=.8 so find y'(t), iss y (t) = el este cosett ez e 3t sinat & t ezt 17 192 y' (t) = q size tatt titsina * (*)} 1790? -'sinst * *** *4-4)3 N 90 = 4*+53 +63(4) + 4 + 5% -39 +46 + it tim = 8 -3X869 + 1 + i +40 = 8 17 - 3x17x.869 +17+4) +464 - 172 -153 + 4 2 = 8

4 Ca = 8+253 462 = 261 :eq=20365 [2= 651 Attoor This y(t) = 151 1034 t + os sinatzt & + (1 - - 12 ) ) Here N e since as ati singt ane Bounded: function & t it linearit ezt if exponential take time to so if est o - at so as t os y (t) so