# Diffy Q

I saw the exponential equations thread and it made me think of my friend who always says that differential equations are much harder can somebody explain them to me. differential equations?

an equation involving derivatives of a function or functions.

A differential equation is a mathematical equation that relates some function of one or more variables with its derivatives. Differential equations arise whenever a deterministic relation involving some continuously varying quantities (modeled by functions) and their rates of change in space and/or time (expressed as derivatives) is known or postulated. Because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.

Differential equations are mathematically studied from several different perspectives, mostly concerned with their solutions — the set of functions that satisfy the equation. Only the simplest differential equations are solvable by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form. If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.

The Sultan - A differential equation is a mathematical equation that relates some function of one or more variables with its derivatives. Differential equations arise whenever a deterministic relation involving some continuously varying quantities (modeled by functions) and their rates of change in space and/or time (expressed as derivatives) is known or postulated. Because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.

Differential equations are mathematically studied from several different perspectives, mostly concerned with their solutions — the set of functions that satisfy the equation. Only the simplest differential equations are solvable by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form. If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.
Didn't understand. But thanks! Lol

Well they are like circular problems where one or more of the variables changes with a variable.

So on the surface it looks like a circular reference. Classic example is calculate the speed of an object dropped from a high building.

If this was a vacuum it would be easy as its just gravity. However if you take wind resistance into account it gets more complex as wind resistance goes up as velocity goes up.

So how can you solve how fast(velocity) the object will be moving after one second. You have to calculate the wind resistance but to calculate wind resistance you have to know the velocity what we don't know or that is what we are trying to solve.

I probably explained it poorly but that is the jist of it.

To explain this poorly and I am sure this is flawed as I have not done this since college.

Dropping a ball is gravity (lets say 9.8Meters per Second^2) minus wind resistance(drag)

So in a vacuum after 1 second the velocity will be 9.8 M/S. However on earth we know it would be slightly slower as we need to take drag into account.

To calculate drag (I had to look this up to remember it) is calculated as Fd(drag) = ½?v²ACd. In this example all you need to know is to calculate drag we need to know velocity (that is the v² in the equation)

So now we are fucked right; to calculate the velocity we need to know the drag. To calculate drag we need to know the velocity.

So then you say; well lets estimate; lets assume there is zero drag for the first second, so at second one we will just say the velocity of the ball is 9.8m/s. Then we will measure the drag at 1 second; then assume the drag is constant from second 1 through 2(it really won't be as the ball will still be accelerating). At second 2 we will do the same thing re-calculate velocity; re-calculate drag assume its constant until second 3 and so on.

Well its better than nothing but still won't give you an accurate reading after a few seconds.

So lets say instead of doing it in one second intervals you do it in 0.1; you get a better estimate.

Now take it further and do it at .01 or .001 or .00000001.

A differential equation essentially does this to infinity so you can now get an accurate measure on how fast the ball will be moving.