Back to Module 5

Module 5 Quick Reference

Applications of Second-Order ODEs

Free Vibrations: mx'' + cx' + kx = 0

ω0 = √(k/m)   |   D = c² - 4mk
TypeConditionBehavior
Undampedc = 0cos/sin at ω0
UnderdampedD < 0Decaying oscillation
CriticalD = 0Fastest decay, no osc.
OverdampedD > 0Slow exponential decay

Forced Vibrations

mx'' + cx' + kx = F0cos(ωt)

Key Cases

ω ≠ ω0, c=0: bounded

ω = ω0, c=0: resonance (t sin)

ω ≈ ω0, c=0: beating

c > 0: bounded steady-state

Amplitude = F0/√[(k-mω²)²+(cω)²]

RLC Circuits

Lq'' + Rq' + (1/C)q = E(t)

Key Facts

i(t) = q'(t)

ω0 = 1/√(LC)

Z = √[R²+(ωL-1/(ωC))²]

Classify: R² vs 4L/C

Mechanical-Electrical Analogy

Mech.Elec.Mech.Elec.
m (mass)L (inductance)x (displ.)q (charge)
c (damping)R (resistance)v (velocity)i (current)
k (spring)1/C (elastance)F (force)E (voltage)
Same ODE, same solution, different physical meaning