Stedy State of damped system

1. May 1, 2012

FHamster

1. The problem statement, all variables and given/known data

Find the steady-state solution having the form https://webwork3.math.ucsb.edu/webwork2_files/tmp/equations/e1/348e8eb8a4ddf62dd06b46276196e71.png [Broken] for the damped system x'' + x' + x = 2cos(3t)

2. Relevant equations

Acos3t + bsin3t

3. The attempt at a solution

To be honest, I wasn't sure how to do this problem, so I just tried undetermined coefficients and got (-16/73)cos(3t)+ (6/73)sin(3t), which was wrong :< muuu

Last edited by a moderator: May 6, 2017
2. May 1, 2012

sharks

Why is (-16/73)cos(3t)+ (6/73)sin(3t) less than the variable "muuu"? :surprised

3. May 1, 2012

ehild

It is the correct steady-state solution, but you need to convert it to the given form xss=Ccos(3t-δ).

ehild

Last edited by a moderator: May 6, 2017
4. May 1, 2012

HallsofIvy

Staff Emeritus
$$Acos(\omega t- \delta)= Acos(\delta)cos(\omega t)- Asin(\delta)sin(\omega t)$$
With $\omega= 3$. What are A and $\delta$?

5. May 1, 2012

andrien

yo need to calculate the particular integral of it.
WHICH WILL BE
2cos(3t)/(D^2+D+1)
where D is what I think you can guess.multiply and divide by D^2-D+1 on left.the denominator will contain only even powers of D.put D^2=-9 in denominator and carry out the differentiation in numerator after that to find the result and if you don't get it see any book on differential eqn to find out the P.I. of it.C.F.will not contribute because it will be zero in steady state.