Question:
Consider the equation for free mechanical vibration, my'' + by' + ky=0, and assume the motion is overdamped. Suppose y(0) > 0 and y'(0) > 0. Prove that the mass will never mass throught its equilibrium at any positive time.

Really stuck on this question, all i know is for over damped:
b^2 > 4mk
General solution: y(t)=C1e^(r1t) + C2e^(r2t) --> y(0) = C1 + C2 > 0
y'(t)=C1r1e^(r1t) + C2r2e^(r2t) --> y'(0) = C1r1 + C2r2 > 0

Any help with this question would be really appreciated thanks.

2. See attachment

3. thanks for replying is it possible if u could explain the steps after A-B .. really confusing

4. A + B >0 comes from y(0) > 0

A - B >0 comes from y ' (0) > 0

So A and B can't both be negative further A > B so A > 0

A > -B

1> -B/A

I then set y equal to 0

e^(-bt/2m) > 0

so if y = 0 then Ae^(rt) +Be^(-rt) = 0

multiply by e^rt

A e^(2rt) + B = 0

e^(2rt) = -B/A

take logs

2rt = ln(-B/A) if B s also postive no solutions.

if B < 0 -B/A < 1 from above and ln(x) < 0 if x < 1

hope this clears things up

5. thx for replying ... i finally understood ... btw its ur own solution right ? not from some solution manual ? thx