Exercise
https://texercises.com/exercise/differential-equation-for-damped-oscillation/
Question
Solution
Short
Video
\(\LaTeX\)
No explanation / solution video to this exercise has yet been created.

Visit our YouTube-Channel to see solutions to other exercises.
Don't forget to subscribe to our channel, like the videos and leave comments!
Exercise:
Show that the damped oscillation yt A e^-delta t cosomega t with omega^omega_^-delta^ is the solution to the differential ddot yt -omega_^ yt-deltadot yt with the initial conditions yA and dot y-Adelta.

Solution:
The first and second derivatives are as follows: dot yt Alefte^-delta t-delta cosomega t-e^-delta tsinomega tomegaright A e^-delta tleft-deltacosomega t-omegasinomega tright ddot yt Ae^-delta t-deltaleft-deltacosomega t-omegasinomega tright nonumber & quad +A e^-delta tleft+deltasinomega tomega-omegacosomega tomegaright A e^-delta tleftdelta^-omega^cosomega t+deltaomegasinomega tright For the right-hand side of the differential we find -omega_^ yt-delta dot yt -A e^-delta tleftomega^+delta^cosomega t -deltaleftdeltacosomega t+omegasinomega trightright A e^-delta tleftdelta^-omega^cosomega t+deltaomegasinomega tright This is equal to the left-hand side of the differential so the given function is a solution to the differential . The initial conditions can easily be verified.
Meta Information
\(\LaTeX\)-Code
Exercise:
Show that the damped oscillation yt A e^-delta t cosomega t with omega^omega_^-delta^ is the solution to the differential ddot yt -omega_^ yt-deltadot yt with the initial conditions yA and dot y-Adelta.

Solution:
The first and second derivatives are as follows: dot yt Alefte^-delta t-delta cosomega t-e^-delta tsinomega tomegaright A e^-delta tleft-deltacosomega t-omegasinomega tright ddot yt Ae^-delta t-deltaleft-deltacosomega t-omegasinomega tright nonumber & quad +A e^-delta tleft+deltasinomega tomega-omegacosomega tomegaright A e^-delta tleftdelta^-omega^cosomega t+deltaomegasinomega tright For the right-hand side of the differential we find -omega_^ yt-delta dot yt -A e^-delta tleftomega^+delta^cosomega t -deltaleftdeltacosomega t+omegasinomega trightright A e^-delta tleftdelta^-omega^cosomega t+deltaomegasinomega tright This is equal to the left-hand side of the differential so the given function is a solution to the differential . The initial conditions can easily be verified.
Contained in these collections:

Attributes & Decorations
Branches
Damped Oscillations
Tags
damped oscillation, exponential envelope
Content image
Difficulty
(3, default)
Points
0 (default)
Language
ENG (English)
Type
Show
Creator by
Decoration
File
Link