Second-Order Linear ODEs

A Second-Order Linear ODE takes the form $ay'' + by' + cy = f(x)$ . When the homogeneous case applies ( $f=0$ ), we test $y = e^{rx}$ to find the characteristic equation $ar^2 + br + c = 0$ .

The general solution depends on the discriminant $\Delta = b^2 - 4ac$ :

Case	Roots	General Solution ( $y$ )
$\Delta > 0$	$r_1 \neq r_2$ (real)	$C_1 e^{r_1 x} + C_2 e^{r_2 x}$
$\Delta = 0$	$r$ (repeated)	$(C_1 + C_2 x)e^{rx}$
$\Delta < 0$	$\alpha \pm \beta i$ (complex)	$e^{\alpha x}(C_1\cos\beta x + C_2\sin\beta x)$

For non-homogeneous equations, the complete solution is $y = y_h + y_p$ using undetermined coefficients or variation of parameters.

Placeholder: Second-Order Linear ODEs

Damped Oscillators & Pitfalls

Common pitfall: Stopping at $y_h$ misses the steady-state ( $y_p$ ), while stopping at $y_p$ drops the transient. You must include both.

Adding damping to a spring gives $m\ddot{x} + b\dot{x} + kx = 0$ . Behaviour is governed by the damping ratio $\zeta = \frac{b}{2\sqrt{mk}}$ :

Damping Type	Ratio Condition	Physical Behaviour
Underdamped	$\zeta < 1$	Oscillations with exponential decay
Critically damped	$\zeta = 1$	Fastest return to zero without oscillation
Overdamped	$\zeta > 1$	Slow, sluggish exponential return

Physics link: Critical damping ( $\zeta = 1$ ) is used in door closers and shock absorbers to achieve equilibrium in minimum time.

Damped Oscillator

Damped Oscillators & Pitfalls

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Differential Equations and Vector Calculus