Consider the initial value problem my+cy+ky=F(t), y(0)=0, y(0)=0 modeling the mo
ID: 1720508 • Letter: C
Question
Consider the initial value problem my+cy+ky=F(t), y(0)=0, y(0)=0 modeling the motion of a spring-mass-dashpot system initially at rest and subjected to an applied force F(t), where the unit of force is the Newton (N). Assume that m=2 kilograms, c=8 kilograms per second, k=80 Newtons per meter, and F(t)=40et Newtons. a) Solve the initial value problem. b) Determine the long-term behavior of the system. Is limty(t)=0? If it is, enter zero. If not, enter a function that approximates y(t) for very large positive values of t.
Explanation / Answer
equation is
2y" +8y' + 80y = 40e^(-t)
=>
y" +4y' + 40y = 20e^(-t)
=>
general solution of homogenous equation is
y = c1e^(-2t)sin 6t + c2e^(-2t)cos6t...............(1)
let y = ke^(-t) be a particular solution
=>
ke^(-t) -4ke^(-t) +40ke^(-t) = 20e^(-t)
=>
37k = 20 => k = 20/37
=>
yp = 20e^(-t) /37
=>
general solution of the original equation is
y =c1e^(-2t)sin 6t + c2e^(-2t)cos6t+ 20e^(-t) /37
y(0) = 0
=>
0 = c2+20/37 => c2 = -20/37
y'(0) = 0
=>
0 = -2c2 +6c1 -20/37
=>
c1 = -20/(6*37) = -10/111
=>
y = [-10 e^(-2t)sin 6t - 60e^(-2t)cos6t+ 60e^(-t)] /111