Modeling with differential equations

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Intros
Lessons
  1. What sort of problems can be solved by first-order differential equations?
  2. Modeling Population Dynamics
  3. Logistic Modeling
  4. Newton's Law of Cooling
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Examples
Lessons
  1. Modeling and Solving Differential Equations
    The rate of change of the volume of a spherical balloon in terms of radius is equal to the surface area of the balloon itself. Find the general solution of the differential equation.

    1. You are working in a factory that converts liquid X to liquid Y at a rate that is proportional to the cubic of the amount of X. Initially, you start with 100 grams of liquid X. After 2 hours, you only have 30 grams of liquid X remaining. How much of liquid X is remaining after 4 hours?
      1. Solving and Examining the Behaviour of the General Solution
        You are studying for a math test that is out of 100. Assume that if you do not study, then you will get 30% on your test. The mark on your math test is related to how much you study according to the model

        SdSdt=1S\frac{dS}{dt}=1

        Where SS is the mark on your math test out of 100 and tt is the amount of hours studied.

        a) Solve this differential equation
        b) How much do you have to study in order to get 100 on your math test?
        c) Is it possible to get over 100 on your math test?
        1. Let F(t)F(t) be the number of Facebook friends you have at time tt. Assume that F(t)F(t) satisfies the logistic growth equation:

          dFdt=0.1F(1F100)\frac{dF}{dt}=0.1F(1-\frac{F}{100})

          a) Find F(t)F(t). (do not find the constant)
          b) Find lim\limt →\infty F(t)F(t). What does this result mean?
          1. Population Dynamics
            A population of Nudibranchs grow at a rate that is proportional to their current population. Without any outside factors the population will double every month.
            If the population is originally 100 Nudibranchs, how many Nudibranchs will there be after 3 months?
            1. Logistic Modeling
              A population of Nudibranchs grow at a rate that is proportional to their current population. Without any outside factors the population will double every month. However the ecosystems the Nudibranchs are living in can only support 15,000 Nudibranchs. If the population is originally 100 Nudibranchs, how many Nudibranchs will there be after 3 months?
              1. Newton's Law of Cooling
                Dr. Daniel Pierce is investigating a crime scene. A corpse was found and the temperature was recorded to be 26°C with the surrounding room temperature being 22°C. The temperature of the corpse was measured two hours later and was found to be 24°C. Assuming that prior to being murdered the person was 37°C, how long ago was the murder committed?