An oscillatory motion is in general governed by a differential equation which looks like

 

d²y/dt² + a₁dy/dt + a₂y = M(t)

 

where y(t) : is a parameter whose time evolution is studied: displacement of a pendulum, current or charge in a circuit etc. a₁ and a₂ are positive constants, where a₁ could be zero or non zero. M(t) could be zero or non-zero periodic quantity, like a periodic driving force.

 

•  Simple Harmonic motion (Pendulum -> LC circuit) 

      d²y/dt² + a₂y = 0

 

•  Damped simple Harmonic motion (Pendulum – LCR       circuit)

 d²y/dt² + a₁dy/dt + a₂y = 0

 

•        Forced damped simple Harmonic motion (Resonance in an LCR circuit) 

         d²y/dt² + a₁dy/dt + a₂y = F₀ Sin(ωt)

 

 

•      Coupled Harmonic oscillators (Two LC circuits coupled with a capacitor)

   

 

• Nonlinear harmonic oscillators (Pendulum) 

d²y/dt² + a₂f(y) = 0