Second-order ODEs and oscillations

Why second-order equations appear

Newton's second law connects force to acceleration:

$F = ma$

Acceleration is the second derivative of position:

a = rac{d^2x}{dt^2}

Therefore many mechanics problems lead naturally to second-order differential equations. These equations involve an unknown function and its second derivative.

A second-order ODE requires two initial conditions, usually initial position and initial velocity.

The spring-mass system

Consider a mass attached to an ideal spring. Hooke's law says the spring force is

$F=-kx$

where $x$ is displacement from equilibrium and $k$ is the spring constant. Newton's second law gives

mrac{d^2x}{dt^2}=-kx

Rearrange:

rac{d^2x}{dt^2}+rac{k}{m}x=0

This is the equation of simple harmonic motion.

Form of the solution

The solution is sinusoidal:

$x(t)=Acos(omega t + phi)$

where $A$ is amplitude, $omega$ is angular frequency, and $phi$ is phase constant. For a spring-mass system,

omega = sqrt{rac{k}{m}}

The motion repeats because acceleration is always proportional to displacement but opposite in direction:

$a=-omega^2x$

If the mass is displaced to the right, acceleration points left. If displaced left, acceleration points right. The restoring effect produces oscillation.

Meaning of constants

The amplitude $A$ is the maximum displacement from equilibrium. The angular frequency $omega$ tells how rapidly the oscillation cycles. The phase $phi$ determines where in the cycle the motion begins at $t=0$.

The period is the time for one full cycle:

T=rac{2pi}{omega}

The frequency in cycles per second is

f=rac{1}{T}

and

$omega = 2pi f$

Velocity and acceleration

Differentiating position gives velocity:

$v(t)=-Aomegasin(omega t+phi)$

Differentiating again gives acceleration:

$a(t)=-Aomega^2cos(omega t+phi)$

Since $x(t)=Acos(omega t+phi)$, this confirms

$a(t)=-omega^2x(t)$

Damping and driving

Real oscillators lose energy due to friction, air resistance, or internal effects. This is damping. A damped oscillator has decreasing amplitude over time. If an external periodic force is applied, the oscillator is driven.

When the driving frequency is near the natural frequency, resonance can occur, producing large oscillations. Resonance appears in musical instruments, bridges, buildings, circuits, and atoms.

Beyond springs

Simple harmonic motion appears whenever a system near stable equilibrium has a restoring effect approximately proportional to displacement. Small pendulum swings, vibrating molecules, sound waves, and electrical oscillations can all be modeled using similar equations.

The big idea

Second-order ODEs are natural in mechanics because acceleration is a second derivative. The equation $d^2x/dt^2 + omega^2x=0$ describes simple harmonic motion, where acceleration is proportional to displacement and opposite in direction. Its sinusoidal solutions form the mathematical foundation for waves, vibrations, sound, circuits, and quantum systems.