Question

Both roots of the characteristic are negative. Explain why this signals that all solutions collapse to 0 without oscillation. ​

G.9.c) Damped but not oscillating solutions Here is an unforced linear oscillator diffeq: In[21] b Random[Real, {4, 6}]; c Random[Real, {3, 4}]; startery = Random[Real, {5, 8}]; starteryprime = Random [Real, {-2, 2}]; Clear [t, yl; linoscdiffeq = {y''[t] + b y'[t] + c y[t] linoscdiffeq == startery, y'[0] == starteryprime}; 0, y[0] Out[26]= (3.98634 y[t] + 4.89966 y [t] + y"[t] = 0, y[0] -= 6.62589, y' [0] == 1.35166} The solutions of the characteristic equation are: In[27]:= Clear[z]; charequation = z^2 + b z + c == 0; zsols = Solve [ charequation, z] Out[29]= {{z-3.86945}, {z--1.03021}} Both roots of the characteristic are negative Explain why this signals that all solutions collapse to 0 without oscillation

Answer 1

The given dliftereuta egc yit) +4-899 ( y'ltt 3.336 34 ylt)-O TLe cacarectnistic atin is z2 4.83966Z +3.98634 = 0 7 483 9 (f 4839 6 - CU (7-3 29 יג .0702 , -3.86 34 S .General solution is se . ס 2/ 3פבןL t 3. 86945 i + given ylo) 6.62583 C2583 Ce difterntiahingwin respet to t ti 2- 945 C.e3863 45" t - 1-0302l t 10302/0e given g 1-3 SI6C 1316-3.66995 & 3.8634 54-1.03021 13516 Solvig ad 2-88025 3.50f14 Puttig hee valuy o and , in we set the Solution t 2.88& 025 e3.869 45t9.50C14e perio di functisn qt .Soutisn is th Jcillti. It is not a As t Fit) 2.83 015 e+9.s0614e 0o . Al olutio collapce to o