← Differential Equations

Unit 4: Modeling with First-Order Equations

Growth, cooling, mixing, and falling with drag, all written down as first-order equations and solved with real numbers.

8 topics · 29 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

2.3 Exponential Growth and Decay Models

  1. How it worksHow does the equation dy/dt = ky turn a constant percentage growth rate into an exponential curve?
  2. Worked exampleSolve dy/dt = 0.03y with y(0) = 1000 and find y at t = 10.
  3. Common mistakeWhy doesn't a quantity growing exponentially ever actually reach infinity in finite time?
  4. Going furtherWhat happens to the growth curve when the rate constant k itself slowly drifts over time instead of staying fixed?
Your turn1 of 2
Solve dy/dt = 0.0618y with y(0) = 1473 and find y at t = 19.

2.3 Half-Life and Doubling Time

  1. Worked exampleFind the half-life of a substance that decays according to dy/dt = -0.021y.
  2. Worked exampleFind how long it takes an investment growing at a continuous rate of 5 percent per year to double.
  3. Going furtherWhat happens to a substance's half-life if its decay constant is doubled?
  4. Common mistakeWhy isn't the time to decay from 100 grams to 75 grams the same length as the time to decay from 75 grams to 50 grams?
Your turn1 of 2
Find the half-life of a substance that decays according to dy/dt = -0.0727y.

2.3 The Logistic Model of Population Growth

  1. How it worksHow does the factor (1 - P/K) in the logistic equation choke off growth as P approaches the carrying capacity K?
  2. Worked exampleSolve dP/dt = 0.4P(1 - P/500) with P(0) = 50 for P as a function of t.
  3. Common mistakeWhy doesn't a population modeled by the logistic equation grow exponentially forever the way unlimited growth would?
Your turn1 of 2
Solve dP/dt = 0.479P(1 - P/460) with P(0) = 76. Find P(t).

2.3 Setting Up a Mixing Problem

  1. How it worksHow do you build the rate-in-minus-rate-out balance that becomes a tank's differential equation?
  2. How it worksHow does a changing tank volume complicate the rate-out term when the inflow and outflow rates differ?
  3. Going furtherWhat happens to a mixing problem's equation once the tank starts overflowing at a fixed maximum volume?
  4. Worked exampleFind the differential equation for a tank starting with 50 gallons of pure water as brine at 2 lb/gal flows in at 4 gal/min with no outflow.
Your turn1 of 2
A tank starts with 65 gallons of pure water. Brine at 3 lb/gal flows in at 6 gal/min with no outflow. Find dy/dt (the rate salt enters the tank, in lb/min) at t = 0.

2.3 Solving a Tank Mixing Problem

  1. Worked exampleFind y(t), the salt in a 100 gallon tank starting with 20 pounds, as 0.5 lb/gal brine flows in and drains out at 3 gal/min.
  2. Worked exampleFind how much salt is in the tank from the previous setup after 30 minutes have passed.
  3. Common mistakeWhy isn't the salt concentration inside the tank perfectly uniform the instant new brine pours in?
Your turn1 of 2
Find y(t), the salt (in pounds) in a 120 gallon tank starting with 29 pounds, as 0.759 lb/gal brine flows in and drains out at 3 gal/min.

2.3 Newton's Law of Cooling

  1. How it worksHow does the rate dT/dt = -k(T - Tambient) make the cooling rate proportional to the temperature gap?
  2. Worked exampleFind the temperature after 20 minutes of coffee that cools from 90 to 70 degrees C in 10 minutes in a 20 degree C room.
  3. Going furtherWhat happens to a cooling curve when the surrounding room temperature is drifting instead of staying fixed?
  4. Common mistakeWhy doesn't an object's cooling rate depend on how much of it there is, only on the gap to room temperature?
Your turn1 of 2
Coffee cools from 92 to 70 degrees C in 9 minutes in a 24 degree C room. Find its temperature after 16 minutes.

2.3 A Falling Body with Air Resistance

  1. How it worksHow does adding a drag term proportional to velocity turn free fall's constant acceleration into a leveling-off curve?
  2. Worked exampleFind the velocity as a function of time and the terminal velocity of a 70 kg skydiver falling with drag force 10v newtons.
  3. Common mistakeWhy doesn't a falling object with air resistance ever mathematically reach its exact terminal velocity?
  4. Going furtherWhat happens to the terminal velocity when the drag force is proportional to v2 instead of v?
Your turn1 of 2
Find the terminal velocity of a 78 kg skydiver falling with drag force 11v newtons. Use g = 9.8.

2.3 Comparing Growth, Decay, and Saturation Models

  1. How it worksHow does the sign and structure of the rate term decide whether a model grows forever, decays away, or levels off?
  2. Worked exampleFind whether dy/dt = -0.2y, dy/dt = 0.2y(1 - y/10), and dy/dt = 5 - 0.2y are growth, decay, or saturating models.
  3. Ties togetherHow does a saturating model like terminal velocity connect to the stable equilibrium you'd find on a phase line?
Your turn1 of 2
A saturating model dy/dt = 38 - 4y has one equilibrium. Find its value.