Modeling with First-Order Equations

Differential equations pervade the sciences because a statement about a rate of change is a statement about a derivative. Newton's cooling, radioactive decay, and population growth are each an assertion that some quantity changes at a rate proportional to something measurable, and each becomes a first-order equation the moment it is written in symbols. Modeling proceeds in three steps.¹

• Construct. Translate the governing physical principle into an equation relating a quantity and its derivative, stating clearly what law is assumed.
• Analyze. Solve the equation, or extract qualitative information about its solutions when a closed form is out of reach.
• Compare. Interpret the solution in context and test it against observation; a model is validated by its predictions, not its derivation.

The equations are approximate by construction: insect populations do not grow without bound, heat transfer depends on more than temperature difference, and no tank is perfectly stirred. Knowing where a model breaks down is part of using it.

Mixing

A tank holds $100$ gallons of water with $Q_0$ pounds of salt dissolved at time $t=0$. Brine at concentration $\frac{1}{4}$ lb/gal flows in at $r$ gal/min, and the well-stirred mixture drains at the same rate. The governing principle is conservation of salt: the rate of change of salt equals the rate in minus the rate out.

Salt enters at $\frac{r}{4}$ lb/min and leaves at $\frac{rQ}{100}$ lb/min, so

This is linear. In standard form $Q^{\prime }+\frac{r}{100}Q=\frac{r}{4}$, the integrating factor is $e^{rt/100}$, and the general solution is $Q=25+c{e}^{-rt/100}$. Applying $Q(0)=Q_0$,

The limiting amount is $Q_L=25$ lb, obtainable without solving by setting $Q^{\prime }=0$: eventually the tank contents match the inflow concentration, $\frac{1}{4}\times 100=25$ lb. The exponential records how the initial charge $Q_0$ relaxes to that steady state.

The same tank model, with a time-varying inflow concentration, produces an oscillating steady state. If chemical enters a pond at concentration $2+\sin (2t)$, the content settles into a fixed oscillation about a constant level once the exponential transient decays — the same transient-plus-steady-state split seen for linear equations.

Exponential growth and decay

When a quantity changes at a rate proportional to its current size, $y^{\prime }=ky$, the solution is $y=y_0{e}^{kt}$: growth for $k>0$, decay for $k<0$. Three standard applications share this skeleton.

• Compound interest. A balance earning return $r$ continuously satisfies $S^{\prime }=rS$, so $S(t)=S_0{e}^{rt}$. With continuous deposits at rate $k$ the equation becomes $S^{\prime }-rS=k$, solved by $S(t)=S_0{e}^{rt}+\frac{k}{r}(e^{rt}-1)$.
• Radioactive decay. An amount $Q$ of a radioisotope obeys $Q^{\prime }=-rQ$, so $Q(t)=Q_0{e}^{-rt}$. The half-life $\tau$ satisfies $e^{-r\tau }=\frac{1}{2}$, giving $r=\ln 2/\tau$. Radiocarbon dating inverts this: measuring $Q/Q_0$ recovers the elapsed time $t=-\frac{1}{r}\ln (Q/Q_0)$.
• Newton's cooling. A body at temperature $u$ in surroundings at $T$ cools at a rate proportional to the difference, $u^{\prime }=-k(u-T)$, so $u(t)=T+(u_0-T)e^{-kt}$ approaches ambient exponentially.

Continuous compounding is the limit of discrete compounding. Compounding $m$ times a year gives $S_0(1+r/m{)}^{mt}$, and ${\lim }_{m\to \infty }(1+r/m{)}^{mt}=e^{rt}$ recovers the continuous model; the frequency of compounding barely matters in practice.

Falling body with drag

A body of mass $m$ falls under gravity against a resistive force proportional to its speed. Taking velocity $v$ positive downward, Newton's second law gives

with $\gamma$ the drag coefficient. In standard form $v^{\prime }+\frac{\gamma }{m}v=g$, this is linear with integrating factor $e^{\gamma t/m}$, and

The exponential decays, so every solution approaches the terminal velocity$v_{\infty }=mg/\gamma$, the speed at which drag exactly balances gravity and $v^{\prime }=0$. The same value falls out of the equation by setting the right side to zero without integrating.

If instead the field varies with altitude, as for a body projected far from the earth, the equation is nonlinear. Writing the gravitational force as $-mg{R}^2/(R+x{)}^2$ and using $\frac{\mathbf{d}v}{\mathbf{d}t}=v\frac{\mathbf{d}v}{\mathbf{d}x}$ to trade time for altitude gives the separable equation $v\frac{\mathbf{d}v}{\mathbf{d}x}=-g{R}^2/(R+x{)}^2$.

Series circuits

An RL circuit with resistance $R$, inductance $L$, and applied voltage $E(t)$ carries a current $i(t)$. Kirchhoff's voltage law sets the applied voltage equal to the sum of the voltage drops — $Ri$ across the resistor and $L\frac{\mathbf{d}i}{\mathbf{d}t}$ across the inductor:²

With constant $E$ the integrating factor is $e^{Rt/L}$ and

The current climbs toward the steady value $E/R$ set by Ohm's law, with the inductor delaying the approach; the time constant $L/R$ fixes the pace. The RC circuit, with charge $q$ obeying $R{q}^{\prime }+q/C=E$, has the identical linear form and the same transient-plus-steady structure. This shared equation is the basis for the mechanical–electrical analogy developed for second-order systems.

Summary

Each model begins with one governing law and ends as a first-order equation whose long-time behavior is the physically meaningful output.

In every case the steady state is obtained by setting the derivative to zero, and the transient records the approach from the initial condition. A nonlinear rate law, such as logistic growth or a threshold population, produces richer behavior from the same setup.