Applications of Higher Order DEQs

Spring Mass System

$$m{y}^{\prime \prime }+b{y}^{\prime }+ky=f(x)$$

• $b^2-4mk$
  • $<0\to$ under damped (oscillations from trig)
  • $>0\to$ over damped
  • $=0\to$ critically damped

$${\omega }_0=\sqrt{k/m},f(x)=cos(\omega x)$$

• Beats when ${\omega }_0\approx \omega$
• Resonance when ${\omega }_0=\omega$
• Applications
  • Beats: Piano tuning
  • Resonance: Everything else