Prove Critical Damping: x(t)=A+Bt e^(-Beta t)

[eku_girl83]

Show that the equation x(t)=(A+Bt)e^(-Beta*t) is indeed the solution for critical damping by assuming a solution of the form x(t)=y(t)exp(-Beta*t) and determining the function y(t).

Is there a differential equation for the critically damped case that I can substitute x(t) and its appropriate derivatives into to solve for y(t)?? Hints, please! There are no examples like this in the text...

 

[Pyrrhus]

This is more of a ODE problem than a physics one as you have notice.
The differential equation for the free damping idealized spring:
$$m \ddot{x} + c \dot{x} + kx = 0$$
For the case of critical damping the characteristic polynomial for this linear ODE indicates repeated roots. Well give it a try.