Steady State Solution for Damped System with External Forcing

[FHamster]

Homework Statement

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

Homework Equations

Acos3t + bsin3t

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

 

[DryRun]

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

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

 

[ehild]

Homework Statement

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

I just tried undetermined coefficients and got (-16/73)cos(3t)+ (6/73)sin(3t), which was wrong :< muuu

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


ehild

 

[HallsofIvy]

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

 

[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.