<aside> 👜 The “Stuff” Equation: Rate of Change of Stuff = Inflow Rate - Outflow Rate + Rate of generation - Rate of Consumption

</aside>

Try to identify:

• Things that are constant over time
• Things that depend explicitly on the time t
• Things that depend on the value of the dependent variable

Constant growth rate

A hose sprays water into a pool at a constant rate of 10 litres each minute

• Let $V(t)$ be the volume of water in the pool after $t$ minutes have elasped
• Rate of change of volume = amount of water coming into the pool per minute: $V’(t) =$ Inflow Rate
• Rate is said to be a constant 10 litres at all times $t$, so $V’(t) = 10$

Time dependent outflow rate

Hose sprays water into a pool at a constant rate of 10 litres each minute. Water flows out of the pool at a rate that varies over time

Rate of outflow oscillates sinusoidally between 2 litres/minute and 4 litres/minute with a period of 30 minutes

• $V’(t) =$ Inflow rate - outflow rate

• Inflow rate is constantly 10 litres/minute at all times $t$. Since the outflow rate depends on time, ti is represented by a function of $t$

  • Properties described by $3 + sin(\frac{2 \pi}{30}t)$
  • Oscillates between 2 litres/minute and 4 litres/minute since $1 ≤\sin(\frac{2 \pi}{30}t) < 1$

• We can model $V(t)$ by the differential equation $V’(t) = 10 - (3 + sin(\frac{2 \pi}{30}t))$

  $= 7 - \sin(\frac{2 \pi}{30}t)$

Newton’s Law of Cooling

A cup of coffee cools at a rate (in degrees/minute) proportional to the difference between the ambient temperature of the room and the current temperature of the coffee, with proportionality constant $c$. If the ambient temperature is a constant 20 degrees, write a differential equation modelling the temperature of the coffee cup after $t$ minutes have elapsed.

• Let $H(t)$ be the temperature of the coffee after t minutes
• According to the prompt, the rate of change of temperature equals $c$ times the difference between the ambient temperature of the room and the current temperature of the coffee
  • This does not quite fit into the “stuff” equation model because degrees Celsius don’t just move from one place to another- heat energy is what moves and it follows a conservation law