# Characteristic Equation
## Linear Differential Equations
**The equation determining the solutions of a given $n$th-order differential equation.**
A characteristic equation can only be formed when the differential equation is *linear*, *homogeneous*, and contains *constant* coefficients.
An $n$th-order differential equation of the form
$a_{n}y^{(n)}+a_{n-1}y^{(n-1)}+\cdots+a_{2}y''+a_{1}y'+a_{0}y=0$
has a *characteristic equation* of the form
$a_{n}r^{n}+a_{n-1}r^{n-1}+\cdots+a_{2}r^{2}+a_{1}r+a_{0}=0$
whose roots $r_{1}$, $r_{2}$, $\dots$, $r_{n}$ form the *general solution* of the homogeneous differential equation.
## Digital Electronics
**The equation determining the next state of a [[latch]] or [[flip-flop]].**
Characteristic equations are expressed in terms of the *inputs* and the *present state*.
For example, the characteristic equation of a T flip-flop is $Q_{\text{next}}=T\bar{Q}+\bar{T}{Q}$.
Characteristic equations are used to generate a [[characteristic table]] or an [[excitation table]].