Second Order Linear Equations

A second-order linear equation is simply a rule that ties a function to its slope and to how fast that slope is changing. It's "second order" because the highest derivative in it is the second one, and "linear" because $y$ and its derivatives only appear on their own, times a number, never squared or multiplied together. Equations like this describe anything with inertia and a restoring force, like springs, circuits and swinging doors. In $ay'' + by' + cy = 0$, the pieces are $y''$ (how fast the slope is changing, like acceleration), $y'$ (the slope, like velocity) and $y$ (the value itself, like position), each scaled by a fixed number $a$, $b$ or $c$, and the three have to balance out to zero. It looks heavier than a first-order equation, but one good guess cracks it open; bear with it.

Why exponentials. Think back to first order: $y' = ky$ is solved by $e^{kt}$, because differentiating an exponential just hands back a multiple of it. That's exactly what we want here. If $y$, $y'$ and $y''$ are all multiples of the same function, the three terms in $ay'' + by' + cy$ have a chance to cancel. So try $y = e^{rt}$:

$e^{rt}$ is never zero, so the only way this works is if $ar^2 + br + c = 0$. In other words, $e^{rt}$ is a solution exactly when $r$ is a root. A calculus problem just turned into an algebra problem.

Superposition. For linear homogeneous equations, any combination of solutions is again a solution. That's because derivatives respect sums and constant multiples: plug in $c_1y_1 + c_2y_2$ and the left side splits into $c_1\cdot0 + c_2\cdot0 = 0$. Two independent solutions are enough to build the general solution.

Three cases. The quadratic formula can give you three kinds of roots:

• two distinct real roots,
• a complex pair,
• a repeated root.

Each one behaves differently and gets its own lesson. Notice too that a second-order IVP needs two initial conditions, usually position and velocity. That makes sense: the general solution has two constants to pin down.

A common mistake. It's tempting to stop after finding one exponential, since $e^{rt}$ already satisfies the equation. It isn't enough: an IVP fixes both a starting position and a starting velocity, and $ce^{rt}$ has only one constant to match two numbers. You need two independent solutions.

Springs, pendulums (for small swings), RLC circuits and building sway are all second-order linear equations: inertia plus a restoring force.