Characteristic Equation

A precise guide to characteristic roots for constant-coefficient linear differential equations.

Classify → choose method → apply condition. A differential equation is classified by order, linearity, and separability before a method is selected. The final condition determines constants and a residual check tests the solution.

Precise definition

For a homogeneous linear equation with constant coefficients $a_ny^{(n)}+\cdots+a_1y'+a_0y=0$, substituting $y=e^{rt}$ gives the characteristic polynomial $a_nr^n+\cdots+a_1r+a_0$. Its roots, with multiplicity, determine a fundamental set of exponential, polynomial-exponential, or sinusoidal solutions.

Notation and mathematical language

For a second-order equation $ay''+by'+cy=0$, write $ar^2+br+c=0$. Distinct real roots $r_1,r_2$ give $C_1e^{r_1t}+C_2e^{r_2t}$. A repeated root $r$ gives $(C_1+C_2t)e^{rt}$. Complex roots $\alpha\pm i\beta$ give $e^{\alpha t}(C_1\cos\beta t+C_2\sin\beta t)$.

Conceptual picture

Differentiation acts on $e^{rt}$ by multiplication by $r$, so a polynomial in the derivative operator becomes the same polynomial in $r$. Repeated roots require extra factors of $t$ because a second independent solution cannot be another constant multiple of $e^{rt}$.

Fully worked example

Interpretation and application

Characteristic roots encode dynamics: negative real parts decay, positive real parts grow, and nonzero imaginary parts oscillate. These statements describe the homogeneous linear model. A forcing term changes the particular response but not the roots that govern the complementary solution.