Table of Contents
Connecting Physical Systems and Differential Equations
In this chapter we focus on how real-world physical systems lead to differential equations. The key idea is always the same:
- Identify a physical quantity that changes (position, temperature, charge, etc.).
- Use physical laws to relate its rate of change to the quantity itself and perhaps to other quantities.
- Translate that relationship into a differential equation.
We will not solve all of these equations in detail here; instead we emphasize model building: how to get from a description of a physical situation to a differential equation.
General Modeling Steps
When building a physical model:
- Choose variables.
Decide what quantity (or quantities) you want to describe. Give them names like $x(t)$, $T(t)$, $Q(t)$, etc. - Choose parameters.
Identify constants in the problem (mass, spring constant, resistance, etc.). These will appear as parameters: $m, k, b, R, C$, and so on. - State assumptions.
Typically we idealize: - no air resistance, or linear resistance only,
- small displacements,
- uniform material,
- constant external forces, etc.
These assumptions often simplify the relationships to linear differential equations.
- Use physical laws.
Apply standard laws such as: - Newton’s second law: $F = m a$,
- Hooke’s law for springs: $F = -kx$,
- Ohm’s law: $V = RI$,
- Conservation of energy or charge,
- Heat flow laws (e.g., Newton’s law of cooling).
- Derive the equation.
Combine the laws and assumptions to express an equation that includes derivatives of your variables. - Specify initial / boundary conditions.
Typical examples: - initial position and velocity,
- initial temperature distribution,
- initial charge on a capacitor.
These conditions pick out a specific solution of the differential equation that matches a particular experiment or situation.
Mass–Spring–Dashpot Systems (Mechanical Oscillators)
A basic mechanical model is a mass attached to a spring, sometimes with a damper (dashpot) providing resistance.
Undamped mass–spring system
Consider:
- a mass $m$ on a frictionless surface,
- attached to a linear spring with constant $k$,
- $x(t)$ is displacement from equilibrium (positive to the right).
Assumptions:
- The spring follows Hooke’s law: restoring force $F_s = -kx$.
- No other horizontal forces (no friction, no damping).
Newton’s second law: total force $=$ mass $\times$ acceleration:
$$
m \frac{d^2 x}{dt^2} = -kx.
$$
Rewriting,
$$
m x'' + kx = 0.
$$
This is a second-order linear homogeneous ODE modeling small oscillations of the mass about equilibrium. The physical content:
- The acceleration is proportional and opposite to displacement.
- The solution describes oscillatory motion (simple harmonic motion).
Initial conditions might be:
- $x(0) = x_0$ (initial displacement),
- $x'(0) = v_0$ (initial velocity).
Damped mass–spring system
Now add a dashpot that produces a damping force proportional to velocity:
- Damping force: $F_d = -b x'$, with damping coefficient $b > 0$.
Total force:
$$
m x'' = -kx - b x'.
$$
So,
$$
m x'' + b x' + kx = 0.
$$
This is the standard model for damped oscillations:
- If damping is small, the motion is oscillatory with slowly decreasing amplitude.
- If damping is large, the system may return to equilibrium without oscillating.
Initial conditions are again values of $x$ and $x'$ at some starting time.
Forced oscillations
Add an external time-dependent force $F_{\text{ext}}(t)$, for example a periodic forcing $F_0 \cos(\omega t)$:
$$
m x'' + b x' + kx = F_{\text{ext}}(t).
$$
For $F_{\text{ext}}(t) = F_0 \cos(\omega t)$:
$$
m x'' + b x' + kx = F_0 \cos(\omega t).
$$
This is a nonhomogeneous ODE modeling a driven oscillator. Phenomena such as resonance emerge from such models (studied more deeply when solving these equations).
Newton’s Law of Cooling (Temperature Models)
Newton’s law of cooling is a simple model for the temperature of an object in a surrounding medium.
Let:
- $T(t)$ be the temperature of the object at time $t$,
- $T_s$ be the constant temperature of the surroundings,
- $k > 0$ be a constant depending on properties of the object and environment.
The law states:
The rate of change of the object’s temperature is proportional to the difference between its temperature and the surroundings.
This gives:
$$
\frac{dT}{dt} = -k \bigl(T - T_s\bigr).
$$
Key features:
- If $T > T_s$, then $T - T_s > 0$, so $dT/dt < 0$: the object cools.
- If $T < T_s$, then $T - T_s < 0$, so $dT/dt > 0$: the object warms.
Initial condition: $T(0) = T_0$ (initial temperature).
This simple model is used in:
- cooling/heating of objects,
- certain forensic time-of-death estimates,
- some elementary engineering heat transfer examples (when assumptions hold).
Electrical Circuits: RC and RLC Models
Many circuits can be modeled by differential equations using Kirchhoff’s laws. Here we look at time-dependent behavior, not static circuits.
RC circuit (charging and discharging a capacitor)
Consider:
- A resistor $R$,
- A capacitor $C$,
- A voltage source $E(t)$ (possibly constant $E_0$),
- $Q(t)$ is the charge on the capacitor at time $t$,
- Current $I(t) = Q'(t)$.
For a simple series RC circuit:
- Voltage across resistor: $V_R = RI$.
- Voltage across capacitor: $V_C = \dfrac{Q}{C}$.
Kirchhoff’s loop rule: sum of voltage drops equals supplied voltage:
$$
RI + \frac{Q}{C} = E(t).
$$
Since $I = Q'$, we get:
$$
R Q'(t) + \frac{1}{C} Q(t) = E(t).
$$
This is a first-order linear ODE for the charge $Q(t)$.
Special case: Constant source $E(t) = E_0$:
$$
R Q' + \frac{1}{C} Q = E_0.
$$
Typical initial condition: $Q(0) = Q_0$ (initial charge, often $0$).
RLC circuit (electrical oscillators)
Now include an inductor with inductance $L$.
In a series RLC circuit:
- Inductor voltage: $V_L = L I'$ (with $I' = dI/dt$),
- Resistor voltage: $V_R = RI$,
- Capacitor voltage: $V_C = \frac{Q}{C}$,
- And again $I = Q'$.
Kirchhoff’s loop rule:
$$
L I' + RI + \frac{Q}{C} = E(t).
$$
Replace $I$ by $Q'$:
$$
L Q'' + R Q' + \frac{1}{C} Q = E(t).
$$
This is a second-order linear ODE for the charge $Q(t)$.
Analogy:
- $L$ plays a role like mass $m$ in the mass–spring model,
- $R$ is analogous to damping coefficient $b$,
- $1/C$ is analogous to spring constant $k$.
Thus an RLC circuit is an electrical oscillator, with behavior similar to a mechanical mass–spring–dashpot system (damped or driven oscillations, resonance, etc.).
Falling Bodies with Air Resistance
The motion of an object falling through a fluid (like air) is affected by drag forces. A simple first model uses linear drag for low speeds.
Let:
- $v(t)$ be the velocity of the object (downward positive),
- $m$ be the mass,
- $g$ be the gravitational acceleration,
- $k > 0$ be the (linear) drag coefficient.
Forces:
- Gravity: $mg$ (downward),
- Drag: $-kv$ (upward, opposite to motion if $v>0$).
Newton’s second law:
$$
m \frac{dv}{dt} = mg - kv.
$$
So,
$$
\frac{dv}{dt} = g - \frac{k}{m} v.
$$
This is a first-order linear ODE.
Key consequences of the model (seen when solving the equation):
- As $t \to \infty$, velocity approaches a constant terminal velocity $v_T = \dfrac{mg}{k}$.
- The object does not accelerate indefinitely; drag balances weight at terminal speed.
Initial condition: $v(0) = v_0$, for example $v_0 = 0$ if dropped from rest.
Simple Population Dynamics as Physical Models
While “population models” have their own subsection, it is useful to note that some of them originate in physical-type conservation ideas: rate of change equals “inflow minus outflow” or “births minus deaths.” A typical structure:
- Let $P(t)$ be a population (or mass, amount of a substance).
- Model:
$$
\frac{dP}{dt} = \text{(rate in)} - \text{(rate out)}.
$$
For example, with:
- constant inflow rate $a$,
- rate of removal proportional to current amount: $b P$,
we get:
$$
\frac{dP}{dt} = a - bP.
$$
This is structurally similar to many physical models:
- heating/cooling with heat input and loss,
- tanks with inflow/outflow of fluid or pollutants,
- chemical reactors with inflow and outflow streams.
Full exploration of population and related models appears in the dedicated “Population models” section, but the physical principle—balance of rates—is common across many modeling contexts.
Mass Balance in Mixing Problems
A classic physical application uses a tank containing a well-mixed solution.
Let:
- $V$ = volume of liquid in the tank (assumed constant),
- $Q(t)$ = amount of solute (e.g., salt) in the tank at time $t$,
- $c_{\text{in}}$ = concentration of incoming solution,
- $r_{\text{in}}$ = inflow rate (volume per unit time),
- $r_{\text{out}}$ = outflow rate, typically $r_{\text{out}} = r_{\text{in}}$ to keep volume constant.
Then:
- Rate in of solute: $r_{\text{in}} c_{\text{in}}$,
- Concentration in tank at time $t$: $Q(t)/V$,
- Rate out: $r_{\text{out}} \cdot \dfrac{Q(t)}{V}$.
Mass balance:
$$
\frac{dQ}{dt} = \text{rate in} - \text{rate out}
= r_{\text{in}} c_{\text{in}} - r_{\text{out}} \frac{Q}{V}.
$$
This again is a first-order linear ODE:
$$
\frac{dQ}{dt} + \frac{r_{\text{out}}}{V} Q = r_{\text{in}} c_{\text{in}}.
$$
Initial condition: $Q(0) = Q_0$ (initial amount of solute).
Energy and Oscillations: Pendulum Approximation
A simple pendulum is a mass $m$ at the end of a light rigid rod of length $L$, swinging under gravity.
Let:
- $\theta(t)$ be the angle from the vertical (small angles),
- Gravity force component tangential to the arc: $-mg \sin\theta$,
- Tangential acceleration: $L \theta''$ (since $s = L\theta$ along the arc).
Newton’s second law in tangential direction:
$$
m L \theta'' = - m g \sin\theta,
$$
so
$$
\theta'' + \frac{g}{L} \sin\theta = 0.
$$
For small angles, use the approximation $\sin\theta \approx \theta$, giving the linearized model:
$$
\theta'' + \frac{g}{L} \theta = 0.
$$
This is exactly the same form as the undamped mass–spring equation, so the small-angle pendulum behaves like a simple harmonic oscillator.
Initial conditions: $\theta(0)$ and $\theta'(0)$ (initial angle and angular velocity).
Summary of Modeling Patterns
Across these physical models, some common patterns appear:
- Second-order linear ODEs from Newton’s second law:
- Mechanical oscillators (mass–spring–dashpot),
- Pendulum (linearized for small angles),
- Electrical RLC circuits.
- First-order linear ODEs from rate balances:
- Newton’s law of cooling,
- RC circuits,
- Falling with linear air resistance,
- Mixing problems,
- Simple population-like models with inflow/outflow.
The same mathematical structures describe very different physical systems. Understanding how to derive these equations from physical principles is the key step in connecting differential equations to the real world.