Uncovering the Practical Applications of Differential Equations

[PrudensOptimus]

What is the true use of Differential Equations?

 

[StarThrower]

Originally posted by PrudensOptimus
What is the true use of Differential Equations?

This question is very vague. The origin of differential equations, was to explain the motion of bodies. If you know the initial conditions, then you also know the future. This was the central idea behind the pioneering work. But to say that predicting the future is the true use of differential equations, is misleading. The true use of differential equations, is this:

Universities can sell knowledge to people, and so make a whole lot of money. That is the true 'use' of differential equations. :)

 

[PrudensOptimus]

Originally posted by StarThrower
This question is very vague. The origin of differential equations, was to explain the motion of bodies. If you know the initial conditions, then you also know the future. This was the central idea behind the pioneering work. But to say that predicting the future is the true use of differential equations, is misleading. The true use of differential equations, is this:

Universities can sell knowledge to people, and so make a whole lot of money. That is the true 'use' of differential equations. :)

OK, I saw a First degree diff eq:


y' + p(x)y = q(x)

as a general solution to y = $$\frac{\int{}u(x)q(x)dx + C}{u(x)}$$

u = exp(∫p(x)dx)


can someone show me what is the "future" in that equation? I thought y' is more of future, and y is initial??

 

[StarThrower]

I am going to just sort of think about the following equation 'aloud' as it were.

y' + p(x)y = q(x)

r(x)dy/dx + r(x)p(x)y = r(x)q(x)

d/dx[ y r(x)] = r(x)dy/dx + ydr/dx

Thus, we need dr/dx = r(x)p(x)

From which it will follow that if we multiply both sides of the original equation by r(x) we have the following sequence of work:

r(x)[y' + p(x)y] = r(x)q(x)
r(x)y' + r(x)p(x)y = r(x)q(x)
d/dx[ y r(x)] = r(x)q(x)

Which will lead us to

d[ y r(x)] = r(x)q(x)dx

Then we can integrate both sides of the above equation to get:

y r(x)= INtegral of [ r(x)q(x)dx ]

And finally we can solve for y(x)

hence the integration factor is found from the following formula:

dr/dx = r(x)p(x)

And all we have to do is solve for r(x), and this is trivial.


dr/r = p(x)dx

From which it follows that



ln (r(x)) = Integral of p(x)

Thus, r(x) equals e^ integral of (p(x)dx)

Or using Latex:

$$ln [r(x)] = \int{}p(x)dx$$

As to how one is to see the future related to the present in this whole thread, all I can say is then when you integrate something over time, you always see how differential equations is intimately connected to determinism.

 

[PrudensOptimus]

Amazing. Give me sometime, I shall cogitate on this matter.

 

[drachman]

What is the use of differential equations?

It is often easy or useful to formulate a problem in terms of a differential equation. Some examples:Radioactive decay--the counting rate in a system of radioactive atoms is proportional to the number of atoms present. The differential equation is dn/dt=-g n, where the derivative is the number of atoms that decay per unit time and g is related to the lifetime. This was trivial to set up, and it has a simple exponential solution. Or consider Newton's famous equation F=ma. Here a is really the second derivative with respect to time:F=m d^2x/dt^2 (for the simple case of one-dimensional motion. In this case two initial conditions are neede, since the equation is of second order:initial position and initial velocity. The problem is then trying to solve the DE. And that keeps us employed!