Characteristic Equation

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 $e^{rt}$.

Conditions and key results

The method applies directly to homogeneous linear equations whose coefficients are constant. It does not solve variable-coefficient or nonlinear equations merely by writing a polynomial. Complex roots occur in conjugate pairs for real coefficients, allowing an entirely real solution basis.

A reliable strategy

1. Move all homogeneous terms to one side and verify constant coefficients.
2. Form and factor or solve the characteristic polynomial, recording multiplicities.
3. Translate each root type into the correct independent real solution terms.
4. Apply initial or boundary conditions and verify by substitution into the differential equation.

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.

Common mistakes

Verification and reasonableness

• Substitute each basis function into the homogeneous equation.
• Compute the Wronskian or observe distinct root construction to confirm independence.
• Check initial conditions after solving for constants, including derivative product-rule terms.

Practice

1. Solve the characteristic equation for $y''-5y'+6y=0$.
2. What form corresponds to a double root $r=-1$?
3. What real form corresponds to roots $-2\pm4i$?

Further deduction

Root multiplicity has an operator interpretation. If the differential operator factors as $(D-r)^m$, then $e^{rt},te^{rt},\ldots,t^{m-1}e^{rt}$ all lie in its null space. For an $n$th-order equation, counting these functions across all roots yields exactly $n$ independent homogeneous solutions, matching the number of initial values required by the existence theorem.

Initial stability can often be read without solving constants. For a real second-order polynomial $r^2+br+c$, both roots have negative real part exactly when $b>0$ and $c>0$. If $c<0$, their product is negative and a growing positive root exists. These coefficient tests are limited to the normalized second-order real case; higher-order stability needs stronger criteria.

Complex conjugate roots arise because real-coefficient polynomials conjugate their nonreal zeros. Combining $e^{(\alpha\pm i\beta)t}$ into real sine and cosine modes preserves a real solution basis and makes the roles of decay rate $\alpha$ and angular frequency $\beta$ explicit.

Related topics

• Second-Order Linear Differential Equations
• Mechanical Vibrations
• Systems of Differential Equations