Question

Find the steady state solution of y''+2y'+2y=cos(wt). Also find the w so that the amplitude of the steady state solution is maximal.

Answer 1

u mean to Find the steady state solution of the differential equation WITHOUT determining the exact solution and taking \(t\) to infinity ?

Answer 2

note that the characteristic equation is \(m^2+2m+2=0\) which has roots

\[m=-1\pm i\]
so u have\[y=e^{-t}(A \cos t+B \sin t)+C \sin \omega t+D \cos \omega t\]

Answer 3

so Actually particular solution will be the steady state solution because the rest of general solution goes to zero exponentially as \(t\rightarrow \infty\)

Answer 4

so just detemine the coefficients of particular solution using undetermined coefficients...

Answer 5

anyway after finding \(C\) and \(D\) in terms of \(\omega\) your amplitude will be \[f(\omega)=\sqrt{C^2(\omega)+D^2(\omega)}\]
then u can find \(\omega\) for which \(f(\omega)\) or amplitude is maximum