Modeling with First-Order Differential Equations

1. Law of Exponential Growth and Decay

When the rate of change of a physical quantity $N(t)$ is directly proportional to the amount currently present, it obeys the foundational differential equation:

If $k > 0$, the process represents exponential growth (e.g., unconstrained bacterial colonies, continuous compound interest). If $k < 0$, writing $k = -\lambda$ ($\lambda > 0$) models exponential decay (e.g., radioactive isotopes). The half-life $t_{1/2}$ is the time required for half the original atoms to decay:

2. Newton's Law of Cooling/Warming

Newton's empirical law states that the time rate of heat loss of a body is proportional to the difference in temperatures between the body and its surroundings:

where $T(t)$ is the temperature of the object and $T_m(t)$ is the ambient surrounding temperature. If $T_m$ is constant, this is a separable equation yielding:

If $T_m(t)$ varies dynamically with time (e.g., daily solar oscillations $T_m(t) = T_0 + A \sin(\omega t)$), the equation is solved as a first-order linear ODE via integrating factor $\mu(t) = e^{kt}$.

1. Principle of Mass Balance in Well-Stirred Mixing Tanks

Consider a tank containing volume $V(t)$ of liquid into which a solute (e.g., salt) is pumped. Let $Q(t)$ denote the mass of solute in the tank at time $t$. The conservation of mass requires:

Let fluid enter at volumetric rate $r_{\text{in}}$ with concentration $c_{\text{in}}$, and leave at volumetric rate $r_{\text{out}}$ with uniform concentration $c_{\text{out}}(t) = \frac{Q(t)}{V(t)}$. Then:

This yields the first-order linear differential equation:

2. Logistic Population Dynamics Under Harvesting

The Verhulst logistic growth model accounts for resource limitation with carrying capacity $K$ and intrinsic reproductive rate $r$. When subject to a constant harvesting rate $H$ (e.g., commercial fishing), the governing equation is:

The equilibrium populations are given by the roots of the quadratic RHS:

Bifurcation Analysis:

• If $H < \frac{rK}{4}$: There are two distinct equilibria $P_1^* < P_2^*$. $P_2^*$ is stable (carrying state) while $P_1^*$ is unstable (threshold extinction boundary).
• If $H = \frac{rK}{4}$: A saddle-node bifurcation occurs at the critical harvesting limit $P^* = K/2$.
• If $H > \frac{rK}{4}$: $\frac{dP}{dt} < 0$ for all $P$, causing catastrophic population collapse to zero in finite time.

1. Motion of a Falling Body Under Drag

By Newton's second law, $m \frac{dv}{dt} = \sum F = m g - F_{\text{drag}}$.

• Linear Drag (Stokes Regime, Low Reynolds Number): $F_{\text{drag}} = k v$. $$\frac{dv}{dt} = g - \frac{k}{m} v \implies v(t) = \frac{m g}{k}\left(1 - e^{-(k/m)t}\right) + v_0 e^{-(k/m)t}$$ The terminal velocity is $v_T = \lim_{t\to\infty} v(t) = \frac{mg}{k}$.
• Quadratic Drag (Newtonian Regime, High Reynolds Number): $F_{\text{drag}} = k v^2$. $$\frac{dv}{dt} = g - \frac{k}{m} v^2 = g\left(1 - \frac{v^2}{v_T^2}\right), \quad \text{where } v_T = \sqrt{\frac{mg}{k}}$$ Separating variables: $\int \frac{dv}{v_T^2 - v^2} = \frac{g}{v_T^2} dt \implies v(t) = v_T \tanh\left(\frac{g t}{v_T}\right)$ (for $v(0) = 0$).

2. Theory of Orthogonal Trajectories

An orthogonal trajectory is a curve that intersects every member of a given family of curves at right angles ($90^\circ$).

• Cartesian Coordinates: Given a family $F(x, y, C) = 0$, differentiate to find the differential equation $\frac{dy}{dx} = f(x, y)$. Since the product of perpendicular slopes is $-1$, the differential equation of the orthogonal family is: $$\left(\frac{dy}{dx}\right)_{\text{orth}} = -\frac{1}{f(x, y)}$$
• Polar Coordinates: Given a family $F(r, \theta, C) = 0$, differentiate to find $r \frac{d\theta}{dr} = f(r, \theta)$. The orthogonal family satisfies: $$\left(r \frac{d\theta}{dr}\right)_{\text{orth}} = -\frac{1}{f(r, \theta)} \implies \left(\frac{dr}{d\theta}\right)_{\text{orth}} = -r^2 \left(\frac{d\theta}{dr}\right)_{\text{orig}}$$