Applications of First-Order DEs

First-order differential equations are the backbone of mathematical modeling in science and engineering. They describe phenomena where the rate of change of a quantity is proportional to the quantity itself or some function of it.

Common Models

  • Growth and Decay: dPdt=kP\frac{dP}{dt} = kP
    • P(t)=P0ektP(t) = P_0 e^{kt}
    • If k>0k > 0: Exponential Growth (Populations, Investments)
    • If k<0k < 0: Exponential Decay (Radioactive Decay, Drug Elimination)
  • Continuous Compound Interest: dAdt=rA\frac{dA}{dt} = rA
    • A(t)A(t): Amount of money at time tt
    • rr: Annual interest rate (continuous)
    • Solution: A(t)=A0ertA(t) = A_0 e^{rt}
  • Newton's Law of Cooling: dTdt=−k(T−Ts)\frac{dT}{dt} = -k(T - T_s)
    • T(t)T(t): Temperature of the object at time tt
    • TsT_s: Constant temperature of the surroundings (ambient)
    • kk: Cooling constant (k>0k > 0)
  • Mixing Problems: dAdt=RinCin−RoutCout\frac{dA}{dt} = R_{in}C_{in} - R_{out}C_{out}
    • A(t)A(t): Amount of solute in the tank at time tt
    • RR: Flow rate (volume/time)
    • CC: Concentration (mass/volume)

Mixing Problems Simulator

Visualize how the amount of solute in a tank changes over time with inflow and outflow.

Mixing Problem Simulator

Parameters

5 L/min
0.5 kg/L
5 L/min
Solute Amount
10.00 kg
Time
0.0 min
Inflow: 5 L/min
Outflow: 5 L/min
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Newton's Law of Cooling Simulator

Visualize how the temperature of an object changes over time as it cools or heats to ambient temperature.

Newton's Law of Cooling

Parameters

100 °C
20 °C
0.05 /min
Time Elapsed
0.0 min
Current Temp
100.0 °C
Ambient
20.0 °C
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Electrical Circuits (RL and RC Series Circuits)

First-order linear differential equations govern simple electrical circuits with constant voltage sources (like a battery, EE) connected in series with resistors (RR) and either inductors (LL) or capacitors (CC).

Circuit Equations (Kirchhoff's Voltage Law)

The sum of voltage drops around a closed loop equals the applied voltage E(t)E(t).
  • RL Series Circuit: The voltage drop across an inductor is LdidtL \frac{di}{dt} and across a resistor is iRiR.
    Ldidt+Ri=E(t)L\frac{di}{dt} + Ri = E(t)
    Where i(t)i(t) is the current.
  • RC Series Circuit: The voltage drop across a resistor is iRiR and across a capacitor is 1Cq\frac{1}{C}q, where q(t)q(t) is the charge. Since i=dqdti = \frac{dq}{dt}, the equation is:
    Rdqdt+1Cq=E(t)R\frac{dq}{dt} + \frac{1}{C}q = E(t)
These are standard linear DEs solvable with an Integrating Factor.

Falling Bodies with Air Resistance

In classic mechanics, a falling body experiences a downward gravitational force (mgmg) and an upward air resistance force proportional to its velocity (kvkv).

Terminal Velocity Model

According to Newton's Second Law (F=maF = ma):
mdvdt=mg−kvm\frac{dv}{dt} = mg - kv
Where mm is mass, gg is acceleration due to gravity, v(t)v(t) is velocity, and kk is a positive constant representing air resistance.
This is a separable and linear differential equation. As t→∞t \rightarrow \infty, the acceleration dvdt\frac{dv}{dt} approaches 0, and the velocity approaches a constant limit known as the terminal velocity, given by vt=mgkv_t = \frac{mg}{k}.

Orthogonal Trajectories

An orthogonal trajectory is a curve that intersects every member of a given family of curves at right angles. It has applications in physics, such as finding electric field lines perpendicular to equipotential curves.

Finding Orthogonal Trajectories

  1. Find the differential equation for the given family of curves: dydx=f(x,y)\frac{dy}{dx} = f(x,y). You may need to eliminate the arbitrary constant parameter first.
  2. Replace dydx\frac{dy}{dx} with the negative reciprocal −dxdy-\frac{dx}{dy} (slope of a perpendicular line).
  3. Solve the new differential equation to find the family of orthogonal trajectories.
Key Takeaways
  • Growth/Decay: Use P(t)=P0ektP(t) = P_0 e^{kt}. kk is positive for growth, negative for decay.
  • Newton's Law of Cooling: Rate of cooling is proportional to temperature difference (T−Tambient)(T - T_{ambient}).
  • Mixing Problems: Balance equation is Rate of Change=Rate In−Rate Out\text{Rate of Change} = \text{Rate In} - \text{Rate Out}. Ensure units match!
  • RL/RC Circuits: Modeled by linear DEs (Ldidt+Ri=EL\frac{di}{dt} + Ri = E or Rdqdt+1Cq=ER\frac{dq}{dt} + \frac{1}{C}q = E) derived from Kirchhoff's Laws.
  • Falling Bodies: Include air resistance proportional to velocity (mdvdt=mg−kvm\frac{dv}{dt} = mg - kv), leading to terminal velocity.
  • Orthogonal Trajectories: Found by solving the DE with the negative reciprocal slope (m⊥=−1/mm_{\perp} = -1/m).
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