A model of a spring/mass system is 4x'' + e−0.1tx = 0. By inspection of the differential equation only, discuss the behavior of the system over a long period of time.

Displacement becomes constant

Explanation:

We know that exponential raised to power infinity (when time is long) equals to zero hence the model system will be as t tends to infinity

4x'' + e−0.1tx = 0

When t tends to infinity

4x'' = 0

Integrating the above equation we get

4x'=c

Integrating it another time we get

4x=cx+d

nah.   i'm just sleepy.