Characteristic Equation

Source lecture(s): PHY621 Lec2, Lec5

Physical Meaning

The characteristic polynomial encodes qualitative behavior of constant-coefficient ODEs.

Variable Definitions

Symbol	Meaning
\(a_n\)	coefficient of \(y^{(n)}\)
\(r\)	root / eigenvalue

Assumptions

Coefficients are constant; trial solution \(y=e^{rx}\) is valid.

Derivation

Substitute \(y=e^{rx}\) into \(a_n y^{(n)}+\cdots+a_0 y=0\), factor out \(e^{rx}\), and cancel.

Applications

Linear systems \(A\vec{v}=\lambda\vec{v}\), RLC stability, quantum harmonic oscillator.

Connections to Other Equations

• eigenvalues-eigenvectors
• ordinary-differential-equations