8.3.4 Modelling with Differential Equations

Modelling with differential equations is a fundamental aspect of applying mathematics to real-world problems. It involves creating equations that describe how a quantity changes over time or with respect to another variable, based on a given set of conditions or assumptions. These models are widely used in fields like physics, biology, economics, and engineering.

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Steps in Modelling with Differential Equations

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Examples of Modelling with Differential Equations

1. Exponential Growth and Decay

Scenario: A population of bacteria grows at a rate proportional to its current size.

• Identify the Variables:
  • Let $P(t)$ be the population at time $t$ .
  • The rate of change of the population is $\frac{dP}{dt}$ .
• Set Up the Relationship:
  • The problem states that the rate of growth is proportional to the population size:

$\frac{dP}{dt} = kP$

where $k$ is a constant of proportionality.

• Formulate the Differential Equation:
  • The differential equation is:

$\frac{dP}{dt} = kP$

• Solve the Differential Equation:
  • This is a separable differential equation:

$\frac{1}{P} \, dP = k \, dt$

• Integrating both sides:

$\ln |P| = kt + C$

• Solving for $P$ :

$P(t) = P_0 e^{kt}$

where $P_0 = e^C$ is the initial population at $t = 0$ .

• Interpret the Solution:
  • If $k > 0$, the population grows exponentially.
  • If $k < 0$ , the population decays exponentially.

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2. Newton's Law of Cooling

Scenario: An object cools in a room at a constant temperature. The rate of change of its temperature is proportional to the difference between its temperature and the room temperature.

• Identify the Variables:
  • Let $T(t)$ be the temperature of the object at time $t$.
  • Let $T_r$ be the room temperature, a constant.
• Set Up the Relationship:
  • According to Newton's Law of Cooling:

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

where $k$ is a positive constant.

• Formulate the Differential Equation:
  • The differential equation is:

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

• Solve the Differential Equation:
  • Separate the variables:

$\frac{1}{T - T_r} \, dT = -k \, dt$

• Integrate both sides:

$\ln |T - T_r| = -kt + C$

• Solve for $T(t)$ :

$T(t) = T_r + (T_0 - T_r)e^{-kt}$

where $T_0 = e^C$ is the initial temperature of the object.

• Interpret the Solution:
  • Over time, the temperature $T(t)$ approaches the room temperature $T_r$ .

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3. Harmonic Motion

Scenario: A mass attached to a spring oscillates back and forth when displaced from its equilibrium position.

• Identify the Variables:
  • Let $x(t)$ be the displacement from the equilibrium position at time $t$ .
  • The force acting on the mass is proportional to the displacement but in the opposite direction (Hooke's Law).
• Set Up the Relationship:
  • Newton's second law states that $F = ma$ , where $a = \frac{d^2x}{dt^2}$ is the acceleration.
  • Hooke's Law gives $F = -kx$ , where $k$ is the spring constant.
• Formulate the Differential Equation:
  • Combining the two:

$m \frac{d^2x}{dt^2} = -kx$

• Simplifying:

$\frac{d^2x}{dt^2} + \frac{k}{m}x = 0$

• Solve the Differential Equation:
  • This is a second-order linear differential equation with constant coefficients.
  • The general solution is:

$x(t) = A \cos(\omega t) + B \sin(\omega t)$

where $\omega = \sqrt{\frac{k}{m}}$ and $A$ and $B$ are constants determined by initial conditions.

• Interpret the Solution:
  • The mass oscillates with angular frequency $\omega$ , and the amplitude and phase depend on the initial conditions.

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Summary