Since the roots are repeated under critically damped we use the general soultion
ID: 3122641 • Letter: S
Question
Since the roots are repeated under critically damped we use the general soultion of x(t) = e^(-ct/2m) * (At + B) where A and B are constants, and we plug it into the equation? I feel like the question is really simple but I haven't faced ODE's since two years ago. Thank you!
(1) Recall the equation describing the spring damped by friction:
m*x''(t) + cx'(t)+kx =0
For this problem, recall the results of Section 13 of our textbook, namely the
formulas (13.1) and (13.2) that give the general solutions of this ODE in the
overdamped and critically damped case.
(a) Consider the critical case c2 = 4mk. Solve the initial value problem with
x(0) = x0 and
x'(0) = v0.
Explanation / Answer
a)x (t) = e^(-ct/2m) *(c1 + c2t).
x(0) =x0
x'(0) = v0
c1 = x0
at t = 0
x'(0) = c2 - c1*c/2m = v0
or c2 = v0 +x0*c/(2m)
hence
x(t) = e^(-ct/(2m))(x0+ (v0 +x0*c/(2m))t)