Given an insight into how something changes, how do you formulate a differential equation that describes it?

Question

I don't mean how do you solve it, or look a differential equation up in a book, I mean how do you formulate one from scratch? Please give an example of how you wold do this.

Answer 1

Describe how the rate of growth depends on the value at that moment. I will give two examples.

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Suppose you have a 1000 gallon tank with salt water. Every second, 1 gallon is pumped out, and 1 gallon of fresh water is pumped in. We are interested how the amount X of salt water changes over time.

Becuase 1/1000 of the water is pumped out every second, we may assume that also 1/1000 of the salt is pumped out (assuming salt is distributed evenly). Therefore, X diminishes at a rate

dX/dt = -X/1000

This is the differential equation describing this situation.

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Suppose X=20 predators and Y=50 prey animals live together. The prey animals breed at a rate of 3% per year. Every year N of the prey animals gets eaten, and the amount N is proportional both to X and to Y, say N = X . Y / 50. For every prey animal eaten, one predator is born; moreover, the predators have a death rate of 5% per year.

To formulate equations, identify the changes in both X and Y:

predators born: N = 0.02 X Y per year
predators die: - 0.05 X per year
prey animals born: +0.03 Y per year
prey animals eaten: -N = -0.02 X Y per year

therefore

dX/dt = 0.02 X Y - 0.05 X
dY/dt = 0.03 Y - 0.02 X Y

This (non-linear) system of differential equations gives a complete description.

Answer 2

Let's take a very simple example: radioactive decay. The assumption is that the number of decays per second is some constant , k, times the amount of the radioactive substance in the sample, multiplied times the amount of time observation is done. SO, let the amount of material at time t be S(t) Then, we set up the equation as:

dS(t) = k*S(t)*dt

Now, that's an awfully simple system, but it gives you the idea. Figure out the things that are factors in the rate of change in the issue of interest; describe them verbally, then turn that description into an equation that expresses the relationship you have set up.

Hope that gives you the general idea of it all. It's really not as hard as it looks.

Answer 3

You must understand the physics well tobe able to formulate problem. Lets take example fo a stone being projected up against gravity. What are the forces acting on the body. Two of them. The inertial force and the gravitational force. acting in oppositedirection. Th last statement is important.
inertial term=mass* acceleration=m*d^2y/dt^2
gravitational force=mass*acceletation due to gravity=m.g
since there are acting in opposite direction we have to have a negative sidn while equting them.
m*d^2/dt^2=-m*g

Answer 4

It really involves when you have a differential relation between items.Take an RC circuit. You know that V=I*R for a resistor and with a capacitor I=C*dv/dt so you would set up an equation. Search for RC circuits or Resistor Capacitor circuits and you should find out how it works.