Second-Order Linear ODEs

$$ay'' + by' + cy = f(x)$$

Homogeneous case ($f=0$) — Try $y = e^{rx}$:

ar^2 + br + c = 0

Three cases (discriminant $\Delta = b^{2}-4ac$):

• $\Delta > 0$, two real roots $r_1 \neq r_2$:

y = C_1 e^{r_1 x} + C_2 e^{r_2 x}

• $\Delta = 0$, repeated root $r$:

y = (C_1 + C_2 x)e^{rx}

• $\Delta < 0$, complex roots $\alpha \pm \beta i$:

$$y = e^{\alpha x}(C_1\cos\beta x + C_2\sin\beta x)$$

Non-homogeneous: $y = y_h + y_p$ (use undetermined coefficients or variation of parameters).

Physics link: These three cases correspond to overdamped, critically damped, and underdamped oscillations.

Common pitfall: For linear ODEs, the full solution is homogeneous + particular. Students who stop at the particular solution lose the transient; those who stop at the homogeneous lose the steady state. You need both.

Damped Harmonic Oscillator

Adding damping to a harmonic oscillator:

$$m\ddot{x} + b\dot{x} + kx = 0$$

The behaviour depends on the damping ratio $\zeta = b/(2\sqrt{mk})$:

• Underdamped ($\zeta < 1$): oscillations with exponential decay
• Critically damped ($\zeta = 1$): fastest return without oscillation
• Overdamped ($\zeta > 1$): slow exponential return

Critical damping is used in door closers, shock absorbers, and galvanometers — it returns to equilibrium in minimum time.