Question

Question 2 < > Solve y"' + 4y' + 8y = 0, y(0) = 1, y'(0) = 6 g(t) = The behavior of the solutions are: O Steady oscillation O Oscillating with decreasing amplitude o Oscillating with increasing amplitude

Answer 1

san: Given that 19"'t Ly'+ By >0, 4107 =), y(0)26 is freemat The dumiliary equation =) mr tamt 8=0 -ut Scuñuxlis m= -4I 16-32 =-2tzi ezt e -zt at1 -u IJ16 autui s The roots of fimi=0 are -2+2i, 2-zi The General solation is yet) = ezt ccicosatt(2 sinat) - g'ct) (-26, sinat +262 cosat) Caicosat +62 sinat) x-ré Griven that 4(o)=1, ylcoat yol=1= y(0)= eoccico Cosot(2 sino) (from -) Cito=1 =) {cial y'0)=6=) ylio) e (-261.70 +262X1) -zeo (atcero) 2C2-201 -26, -6 =) (2-41=3 (2=3t = 3+1 Substitute arcaino (C2=4 The solution is yet) = ezt(coszt ta since) e - 4

Since yct) est cosattuesinat compare yct) with yet) = celt cospt that singt Here 421, C2=4 d=-2, B 20 since అమలు 2<o the behavior of the solution of the given Intial value problem is oscillating with decreasing amplitude