# resonance involing an ODE

• Mar 20th 2010, 12:26 PM
cooltowns
resonance involing an ODE
Hi

my question is

For the ODE below, write down the forcing function $\displaystyle f(t)$that will lead to resonant behaviour $\displaystyle (t\rightarrow\infty)$. Verify your choice by calculating the particular intergral that corresponds to your forcing and examining its limit as $\displaystyle t\rightarrow\infty$
a) $\displaystyle y^{(4)}+5\ddot{y}+4y=f(t)$

my solution to $\displaystyle y^{(4)}+5\ddot{y}+4y=0$ is $\displaystyle y_{c}=Asin(x)+Bcos(x)+Csin(2x)+Dcos(2x)$

thanks
• Mar 21st 2010, 05:55 AM
Jester
Quote:

Originally Posted by cooltowns
Hi

my question is

For the ODE below, write down the forcing function $\displaystyle f(t)$that will lead to resonant behaviour $\displaystyle (t\rightarrow\infty)$. Verify your choice by calculating the particular intergral that corresponds to your forcing and examining its limit as $\displaystyle t\rightarrow\infty$
a) $\displaystyle y^{(4)}+5\ddot{y}+4y=f(t)$

my solution to $\displaystyle y^{(4)}+5\ddot{y}+4y=0$ is $\displaystyle y_{c}=Asin(x)+Bcos(x)+Csin(2x)+Dcos(2x)$

thanks

Ya - if you force your system with an $\displaystyle f(t)$ that contains any member of the complimentary solution it will resonant $\displaystyle {\it i.e.}$ if $\displaystyle f(t) = \sin t$ the solution will contain a terms like $\displaystyle t \sin t$ and $\displaystyle t \cos t$.