Applications of First-Order DEs - Theory & Concepts

Applications of First-Order DEs

Learning Objectives

Formulate and solve first-order differential equations for exponential growth and decay.
Apply Newton's Law of Cooling to find temperature changes over time.
Model and solve mixing problems involving inflow and outflow rates.
Analyze simple RL and RC electrical circuits using Kirchhoff's Voltage Law.
Model falling bodies experiencing air resistance.
Determine the orthogonal trajectories for a given family of curves.

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.

Mathematical Modeling

Many real-world phenomena can be modeled using first-order differential equations. The core idea is to translate the physical principles governing the rate of change of a system into a mathematical equation. These models allow us to predict future states of the system based on its initial conditions.

Exponential Growth and Decay

Models population growth, radioactive decay, and continuously compounded interest.

\frac{dP}{dt} = kP \implies P(t) = P_0 e^{kt}

Variables

Symbol	Description	Unit
$P(t)$	Quantity at time $t$	varies
$t$	Time	varies
$k$	Proportionality constant (growth if $k > 0$ , decay if $k < 0$ )	1/time
$P_0$	Initial quantity at $t = 0$	varies

Newton's Law of Cooling

Models the temperature change of an object in a surrounding medium.

\frac{dT}{dt} = -k(T - T_s)

Variables

Symbol	Description	Unit
$T(t)$	Temperature of the object at time $t$	$^\circ C \text{ or } ^\circ F$
$t$	Time	$s, min, \text{ or } h$
$T_s$	Constant ambient temperature of the surroundings	$^\circ C \text{ or } ^\circ F$
$k$	Cooling constant ( $k > 0$ )	1/time

Mixing Problems

Models the amount of solute in a tank with inflow and outflow.

\frac{dA}{dt} = R_{in}C_{in} - R_{out}C_{out}

Variables

Symbol	Description	Unit
$A(t)$	Amount of solute in the tank at time $t$	$kg \text{ or } lb$
$t$	Time	$s \text{ or } min$
$R_{in}$	Inflow rate	$L/s \text{ or } gal/min$
$C_{in}$	Inflow concentration	$kg/L \text{ or } lb/gal$
$R_{out}$	Outflow rate	$L/s \text{ or } gal/min$
$C_{out}$	Outflow concentration ( $A(t) / V(t)$ )	$kg/L \text{ or } lb/gal$

Interactive Simulation

Visualize how the amount of solute in a tank changes over time with inflow and outflow. Interact with the mixing tank simulation below to see first-order ODEs in application.

Mixing Tank Model

\frac{dA}{dt} = R_{in} C_{in} - R_{out} \frac{A}{V(t)}

Inflow Rate (

R_{in}

)3 L/min

Inflow Concentration (

C_{in}

)2 kg/L

Outflow Rate (

R_{out}

)3 L/min

Salt Amount10.0 kg

Tank Volume100.0 L

A(t) vs t

Electrical Circuits (RL and RC Series Circuits)

First-order linear differential equations govern simple electrical circuits with constant voltage sources (like a battery, $E$ ) connected in series with resistors ( $R$ ) and either inductors ( $L$ ) or capacitors ( $C$ ).

According to Kirchhoff's Voltage Law, the sum of voltage drops around a closed loop equals the applied voltage $E(t)$ . These circuits can be modeled with standard linear DEs solvable with an Integrating Factor.

RL Series Circuit

Models the current in a circuit with a resistor and an inductor.

L\frac{di}{dt} + Ri = E(t)

Variables

Symbol	Description	Unit
$L$	Inductance	H
$i(t)$	Current at time $t$	A
$R$	Resistance	$\Omega$
$E(t)$	Applied voltage	V
$t$	Time	s

RC Series Circuit

Models the charge on a capacitor in a circuit with a resistor and a capacitor.

R\frac{dq}{dt} + \frac{1}{C}q = E(t)

Variables

Symbol	Description	Unit
$R$	Resistance	$\Omega$
$q(t)$	Charge at time $t$	C
$C$	Capacitance	F
$E(t)$	Applied voltage	V
$t$	Time	s

Falling Bodies with Air Resistance

In classical mechanics, a falling body experiences a downward gravitational force and an upward air resistance force. When air resistance is proportional to velocity, this system can be modeled using a first-order differential equation.

Terminal Velocity Model

Applies Newton's Second Law to a falling body experiencing air resistance.

m\frac{dv}{dt} = mg - kv

Variables

Symbol	Description	Unit
$m$	Mass of the body	kg
$v(t)$	Velocity at time $t$	m/s
$t$	Time	s
$g$	Acceleration due to gravity	$m/s^2$
$k$	Air resistance proportionality constant ( $k > 0$ )	kg/s

Terminal Velocity Limit

The terminal velocity model is a separable and linear differential equation. As $t \rightarrow \infty$ , the acceleration $\frac{dv}{dt}$ approaches $0$ , and the velocity approaches a constant limit known as the terminal velocity, given by $v_t = \frac{mg}{k}$ .

Orthogonal Trajectories

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

Finding Orthogonal Trajectories

STEP-BY-STEP

Find the DE: Differentiate the given family of curves to find its differential equation $\frac{dy}{dx} = f(x,y)$ . You may need to eliminate the arbitrary constant parameter first.
Take the Negative Reciprocal: Replace $\frac{dy}{dx}$ with its negative reciprocal $-\frac{dx}{dy}$ (since the slopes of perpendicular lines multiply to $-1$ ).
Solve the New DE: Solve the new differential equation to find the family of orthogonal trajectories.

Key Takeaways

Growth/Decay: Use $P(t) = P_0 e^{kt}$ . $k$ is positive for growth, negative for decay.
Newton's Law of Cooling: Rate of cooling is proportional to the temperature difference $(T - T_s)$ .
Mixing Problems: Balance equation is $\text{Rate of Change} = \text{Rate In} - \text{Rate Out}$ . Ensure units match!
RL/RC Circuits: Modeled by linear DEs derived from Kirchhoff's Voltage Law.
Falling Bodies: Include air resistance proportional to velocity, leading to a terminal velocity.
Orthogonal Trajectories: Found by solving the DE with the negative reciprocal slope ( $m_{\perp} = -1/m$ ).

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