Differential Equations with Applications: First Order ODEs

Applications of First Order Equations

Growth and Decay Problems

• Population growth rate is proportional to the current population: $\frac{dN}{dt} = kN$, where:

  • $N(t)$ is the population size at time $t$.
  • $k$ is a positive constant.

• Law of natural decay: $\frac{dm}{dt} = -km$, where:

  • $m(t)$ is the mass of a radioactive substance at time $t$.
  • $k$ is a positive constant.

Newton’s Law of Cooling

• The rate of change of temperature of a body is proportional to the temperature difference between the body and its surroundings: $\frac{dT}{dt} = k(T - T_m)$, where:
  • $T$ is the temperature of the body at time $t$.
  • $T_m$ is the ambient temperature.
  • $k$ is a constant of proportionality.
  • $\theta = T - T_m$, then $\frac{d\theta}{dt} = k\theta$.

Electrical Problems

• RC Circuit: $RC \frac{dQ}{dt} + Q = 0$, resulting in $Q(t) = Q_0 e^{-\frac{t}{RC}}$, where:

  • $Q$ is the charge in coulombs at time $t$.
  • $R$ is the resistance in ohms.
  • $C$ is the capacitance in Farads.
  • $Q_0$ is the initial charge.

• RL Circuit: $L \frac{di}{dt} + Ri = E$, resulting in $i(t) = \frac{E}{R} (1 - e^{-\frac{Rt}{L}})$, where:

  • $i$ is the current in Amperes at time $t$.
  • $L$ is the inductance in Henries.
  • $E$ is the voltage source.
  • $R$ is the resistance in ohms.