How to solve a second-order linear homogeneous ODE with constant coefficients?

The characteristic equation $ar^2 + br + c = 0$ comes from assuming $y = e^{rx}$. **Why the repeated root case has an extra $x$ factor:** When the characteristic equation has a repeated root $r$, we have one solution $y_1 = e^{rx}$. To find the second linearly independent solution, use reduction of order: Let $y_2 = v(x) y_1$. Substituting into the ODE eventually gives $v'' = 0$, so $v = C_1 + C_2 x$. This gives $y_2 = x e^{rx}$. Intuitively, the solutions are "merging" at the repeated root, and the $x$ factor represents the limit of two distinct exponential solutions as they approach each other. **Physical examples (damped harmonic oscillator):** $$m \frac{d^2x}{dt^2} + c \frac{dx}{dt} + kx = 0$$ - **Overdamped** (distinct real roots): $c^2 > 4mk$, system returns to equilibrium slowly without oscillating - **Critically damped** (repeated root): $c^2 = 4mk$, fastest return to equilibrium without oscillation (ideal for shock absorbers) - **Underdamped** (complex roots): $c^2 < 4mk$, system oscillates with decreasing amplitude The complex case gives $x(t) = e^{-\alpha t}(C_1 \cos \beta t + C_2 \sin \beta t)$, where $\alpha = c/(2m)$ is the damping coefficient and $\beta = \sqrt{4mk - c^2}/(2m)$ is the damped frequency.