Linear ODEs & Exponential Relaxation

The unifying principle

Consider any quantity $x(t)$ whose instantaneous change is proportional to its present value:

$\frac{dx}{dt} = -kx, \qquad k>0.$

Separate variables and integrate:

$\int \frac{dx}{x} = -\int k\,dt \;\Rightarrow\; \ln x = -kt + C \;\Rightarrow\; \boxed{x(t) = x_0\,e^{-kt}}.$

Three structural facts fall out immediately and hold in every field below:

• Time constant: $\tau = 1/k$. At $t=\tau$, $x=x_0/e$.
• Half-life: the time to halve is fixed and independent of $x_0$: $t_{1/2} = \frac{\ln 2}{k} = \tau\ln 2 \approx 0.693\,\tau.$
• Memorylessness: the decay looks identical from any starting point — the process has no "clock" beyond $k$.

For relaxation toward a nonzero equilibrium $x_\infty$, substitute $y = x - x_\infty$; then $\dot y = -ky$ and

$x(t) = x_\infty + (x_0 - x_\infty)e^{-kt}.$

This is the same equation wearing a coordinate shift.

How it shows up in each field

Maths — the linear first-order ODE

The form: $\dot x = -kx$ is the simplest nontrivial linear ODE; $e^{-kt}$ is the eigenfunction of $d/dt$ with eigenvalue $-k$. Solutions form a 1D vector space spanned by $e^{-kt}$; superposition holds. It is the scalar prototype of $\dot{\mathbf x}=A\mathbf x$, whose solution $e^{At}$ decays iff all eigenvalues of $A$ have negative real part.

Physics — RC circuit discharge

The form: a capacitor $C$ discharging through resistor $R$. Kirchhoff gives $Q/C = -R\,\dot Q$, i.e. $\frac{dQ}{dt} = -\frac{1}{RC}\,Q, \qquad k = \frac{1}{RC}.$ Here $\tau = RC$ has units of seconds directly. Same idea: the driving "force" (voltage $Q/C$) pushes current that depletes the charge causing it.

Radioactive decay is the same law with $k=\lambda$ the decay constant: $\dot N=-\lambda N$, each nucleus decaying independently with fixed probability per unit time. Carbon-14: $t_{1/2}=5730$ yr $\Rightarrow \lambda=\ln 2/5730$.

Chemistry — first-order kinetics & drug clearance

The form: a first-order reaction $A\to \text{products}$ has rate $-\dfrac{d[A]}{dt}=k[A]$, giving $[A]=[A]_0e^{-kt}$. In pharmacokinetics, drug elimination is (often) first-order: $\frac{dC}{dt} = -k_e C, \qquad k_e = \frac{\text{CL}}{V_d},$ where CL is clearance and $V_d$ the volume of distribution.

Biology — population decline & neuronal decay

The form: a population with per-capita death rate exceeding birth rate obeys $\dot P = rP$ with $r<0$; writing $r=-k$ gives exponential decline $P=P_0e^{-kt}$. The membrane potential of a leaky neuron relaxes to rest identically: $\tau_m\,\dot V = -(V-V_\text{rest})$, so $V(t)=V_\text{rest}+(V_0-V_\text{rest})e^{-t/\tau_m}$ — the shifted form.

Why this bridge matters

• One measurement, total prediction. Knowing $\tau$ (or $t_{1/2}$) from any domain lets you predict the entire future trajectory — no need to model microscopic detail.
• Cross-field intuition transfer. A chemist's "half-life reasoning" is exactly a physicist's "$5\tau$ settling time" and a pharmacologist's "steady-state after five doses." The $37\%$-per-$\tau$ heuristic is universal.
• Log-linearization diagnostic. Plotting $\ln x$ vs $t$ gives a straight line of slope $-k$ in every field — a shared experimental signature that reveals whether a process is truly first-order.
• The gateway to richer dynamics. Adding a source term $\dot x = -kx + S$ (forced RC, drug infusion, immigration) or coupling ($\dot{\mathbf x}=A\mathbf x$) builds every linear systems theory from this atom.

Connections

• 01 First-Order Linear ODEs
• 02 Eigenvalues and Matrix Exponentials
• 03 RC and RL Circuits
• 04 Radioactive Decay & Half-Life
• 05 First-Order Reaction Kinetics
• 06 Pharmacokinetics & Clearance
• 07 Exponential Population Models
• 08 Leaky Integrate-and-Fire Neuron

#bridge