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Wize University Calculus 2 Textbook > Differential Equations

Applications of DE

Linear Differential EquationsPrevious Section
Second-Order Reducible DEsNext Section
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Common DE Applications

Slope of Tangent Lines

  • These problems involve the slope of a tangent and some points that are on the curve
Wize Concept
Formula: slope of the tangent is dydx\frac{dy}{dx}

Population Growth and Decay

  • These problems involve the increase (growth) or decrease( decay) of a population
  • The population is often given or needs to be found
  • The question can be about half-life (the time it takes for half of the substance to decay)

Wize Concept
Formula: If dPdt=kP\frac{dP}{dt}=kP, then the population at time t is given by P(t)=P(0)ektP\left(t\right)=P\left(0\right)e^{kt}

Newton's Law of Heating and Cooling

  • These problems involve the cooling or heating of a liquid
Wize Concept
Formula: The rate in which the temperature of the liquid is changing is given by dTdt=k(T−Ts)\frac{dT}{dt}=k\left(T-T_s\right) where TsT_s is the temperature of the surrounding

Mixing Tank Problems

  • These problems involve a tank of liquid and some substance that is being added and/or removed from the tank
  • Concentration of a substance is given by amount of substanceamount of liquid\frac{\text{amount of substance}}{\text{amount of liquid}}, usually grams or kilogramsLiters\frac{\text{grams or kilograms}}{\text{Liters}}
Wize Concept
Formula: The rate in which the amount of substance y in the tank is changing is given by dydt=(Ratein)(Concentrationin)−(Rateout)(Concentrationtank) \frac{dy}{dt}=\left(Rate_{in}\right)\left(Concentration_{in}\right)-\left(Rate_{out}\right)\left(Concentration_{\text{tank}}\right)\

Practice Question

The growth rate of a certain species of wild cats at time t years is given by dPdt=kP\frac{dP}{dt}=kP. If we know that the initial population in year 2000 was 500, and that there were double that amount after one and a half years, what is the population of wild cats in the year 2009?

Practice Question

A population is modeled by dPdt=2.5P(1−P3500)\frac{dP}{dt}=2.5P\left(1-\frac{P}{3500}\right)
a.) What are the equilibrium solutions?
b.) For what values of P is the population increasing?
c.) For what values of P is the population decreasing?

Practice Question

Newton’s Law of Cooling is given by dTdt=k(T−M)\frac{dT}{dt}=k\left(T-M\right) .
a. Treating this law as a linear differential equation, what should be the integrating factor?
b. Treating this law as a separable differential equation, what are the two expressions that you need to integrate?

Practice Question

A water tank holds 20L of water. After dissolving 1kg of salt in the tank, a small hole was punctured at the bottom of the tank. A brine solution of 0.2kg of salt per litre is poured into the tank at a constant rate of 2L per minute and the well-stirred mixture leaks out of the tank at the same rate.
Suppose that the change in the amount of salt yy is given by dydt=Ratein×Concentrationin−Rateout×Concentrationtank\frac{dy}{dt}=\text{Rate}_{in}\times\text{Concentration}_{\text{in}}-\text{Rate}_{\text{out}}\times\text{Concentration}_{\text{tank}}
a. Find a formula y(t)y\left(t\right) for the amount of salt in the tank at time t.
b. How much salt is in the tank at 10 minutes?
c. In the long run, how much salt will be in the tank?

Practice Question

A tank initially contains 20 kg of salt and 100 L of pure water. Pure water flows into the tank at a rate of 2 L/min and salt is added to the tank at 0.1 kg/L. The contents of the tank are continuously mixed so that the solution is consistent throughout. The well-mixed solution flows out of the tank at a rate of 3 L/min. How much salt will there be in the tank about 90 minutes?