Characteristic Roots

Definition

The characteristic equation of $a\ddot{x}+b\dot{x}+cx=0$ is the quadratic

$$a{r}^2+br+c=0,r_{\pm }=\frac{-b\pm \sqrt{b^2-4ac}}{2a}.$$

The nature of the roots determines the form of the general solution.

Intuition

The three cases correspond directly to physical behaviour: two distinct real roots give pure exponential growth/decay (overdamped); complex conjugate roots give oscillations wrapped in an exponential envelope (underdamped); a repeated root is the critical transition between the two (critically damped). Euler’s formula $e^{i\mu t}=\cos \mu t+i\sin \mu t$ is the bridge from complex roots to real oscillatory solutions.

Formal Description

Case 1 — Distinct Real Roots ($b^2-4ac>0$)

Two distinct real roots $r_1\ne r_2$ give two independent exponential solutions. The general solution is

$$x(t)=c_1{e}^{r_{1}t}+c_2{e}^{r_{2}t}.$$

Case 2 — Complex Conjugate Roots ($b^2-4ac<0$)

Write the roots as $r=\lambda +i\mu$ and $\bar{r}=\lambda -i\mu$. Using Euler’s formula $e^{i\mu t}=\cos \mu t+i\sin \mu t$, two real independent solutions are

$$x_1(t)=e^{\lambda t}\cos \mu t,x_2(t)=e^{\lambda t}\sin \mu t.$$

The general solution is

$$x(t)=e^{\lambda t}\left(A\cos \mu t+B\sin \mu t\right).$$

The real part of the roots appears in the exponential envelope; the imaginary part appears in the oscillatory terms.

Case 3 — Repeated Root ($b^2-4ac=0$)

The ansatz yields only one independent solution $e^{rt}$. Taking the limit $\mu \to 0$ of the complex-conjugate solution (using ${\lim }_{\mu \to 0}{\mu }^{-1}\sin \mu t=t$) reveals a second independent solution $t{e}^{rt}$. The general solution is

$$x(t)=\left(c_1+c_{2}t\right)e^{rt}.$$

The repeated root $r$ satisfies $r=-b/(2a)$.

Key Results

• All three cases follow from the single exponential ansatz $x=e^{rt}$.
• The principle of superposition justifies forming general solutions as linear combinations.
• Independence of the two solutions is confirmed by the Wronskian being nonzero.

Applications

• Directly gives the transient response of any linear constant-coefficient system.
• The eigenvalue analogue underpins linear ODE systems.
• Stability analysis: the system is stable if and only if all roots have strictly negative real part.

Trade-offs

• Only applies to constant-coefficient equations; variable-coefficient ODEs require other approaches.
• For complex roots, Euler’s formula is required to recover real-valued solutions from complex exponentials.
• The repeated-root case requires a separate argument (limit or reduction of order) to find the second independent solution $t{e}^{rt}$.

Links

• Homogeneous Second-Order ODE
• Principle of Superposition
• Wronskian
• Linear ODE Systems (analogous eigenvalue cases)