growth_decay_oscillation

Growth, Decay, and Oscillation

Definition

Three fundamental behaviours — linear growth, exponential growth/decay, and oscillation — each correspond to a simple ODE. The order of a differential equation is the highest-order derivative it contains.

Intuition

These three ODEs are the simplest non-trivial differential equations and serve as building blocks for understanding more complex systems. Exponential growth arises whenever the rate of change is proportional to the current value; oscillation arises when a restoring force is proportional to displacement.

Formal Description

Linear growth:

$$x(t)=at+b⟺\ddot{x}=0,x(0)=b,\dot{x}(0)=a.$$

Exponential growth ($r>0$) / decay ($r<0$):

$$x(t)=A{e}^{rt}⟺\dot{x}=rx,x(0)=A.$$

Oscillation:

$$x(t)=A\cos \omega t+B\sin \omega t⟺\ddot{x}+{\omega }^{2}x=0.$$

Initial conditions select the specific solution:

• $x(0)=A$, $\dot{x}(0)=0$ $\Rightarrow$ $x(t)=A\cos \omega t$.
• $x(0)=0$, $\dot{x}(0)=\omega B$ $\Rightarrow$ $x(t)=B\sin \omega t$.

Applications

Exponential growth models population growth and compound interest; exponential decay models radioactive decay, cooling, and discharge of a capacitor. Oscillation describes mass-spring systems, pendulums (small angles), and LC circuits.

Trade-offs

These are idealised models: real systems rarely exhibit pure exponential growth indefinitely (resource limits apply) or pure undamped oscillation (friction dissipates energy). More realistic models combine these behaviours, e.g. damped oscillation $\ddot{x}+2\gamma \dot{x}+{\omega }^{2}x=0$.

Links

• Trigonometric Functions — $\cos$ and $\sin$ as periodic solutions
• Natural Logarithm — exponential function and its inverse
• Position, Velocity and Acceleration — physical interpretation of $\dot{x}$ and $\ddot{x}$