Second-order linear equations with constant coefficients

The equations $ay'' + by' + cy = 0$ are solved by seeking solutions of the form $y=e^{\lambda x}$: this leads to the characteristic equation $a\lambda^2 + b\lambda + c = 0$, whose discriminant distinguishes three cases. These are the equations that govern the harmonic oscillator and its physical cousins: the mass-spring (with and without friction) and the RC and RLC circuits. The same mathematical object describes very different phenomena — it is the power of mathematical language.

• The characteristic equation and the three cases
• Harmonic oscillator
• Mass-spring without friction
• Mass-spring with friction
• Discharging RC circuit
• RLC circuit in free oscillation
• The mechanical-electrical analogy