Roger's Equations

Please do show how these functions relate to real world applications of differential equations. Its been over 40 years since I dealt with differential equations in college, and didn't really understand them then.

I have done as you requested. I have only provided examples for some, I will try to give the other functions examples. If its any help, differential equations are to functions what algebra is to numbers. In algebra, you try to find the number that satisfies the equation.

If x + 3 = 9, what is x?

x= 9-3 = 6

Similarly in differential equations, you try to find the function (equation) that satisfies the differential equation.

If dx/dy + 3 = 9, what is x?

dx/dy = 9-3 = 6
∫dx= ∫6dy
x-x₀ = (y-y₀) 6
x = x₀ + 6(y - y₀)

If you look carefully at the solution, you'll see it's just a line (remember x0 and y0 are just "initial conditions" ie constants)

x = 6y + (x₀ - 6y₀)
x = ay + b (equation of a line)

In fact, the point is that ANY time you have the differential equation:

dx/dy = K= Constant;

The solution will be a line

x= Ky + b

What if the differential equation is,

d²x/dy²= K = Constant

then the solution is

x= 1/2Ky² + by + c (quadratic equation)

If the differential equation is

d³x/dy³ = K = Constant

then the solution is

x = 1/6Ky³ + by² + cy + d (cubic equation)

hopefully you're starting to see a general solution emerging in the equations above.

The point I'm trying to make is that each differential equation type has a solution. Now you may be thinking, "but there are so many possible types of differential equations", that's true in MATH, but in physics and engineering, the same old differential equations tend to pop up over and over and over again. If you memorize the solutions of 5 to 10 particular types of differential equations, you're well equipped to most engineering and physics problems.

For instance, if some tells you that:

dN/dt = -λN

then you know that N decreases (decreases because of the sign on lamda) exponentially in time. Notice you didn't need to solve it with calculus, you just needed to remember that differential equations of the form:

dx/dy = Kx

where k is constant, has solution:

x = Ae^-Ky

If you memorize that fact, you don't have to worry about the calculus unless you need to know exact numbers.

The point is, we are forced to memorize all kinds of things like trig identies, derivatives, multiplication tables, but all of a sudden we can't memorize differential equation solutions? Why not?

To answer your last question, I believe because we really didn't understand either the original equations or know of uses for them.

You've given me a good start that I should have had 40 years ago! BTW I commonly tend to skip the equations when reading similar text, but I read (and I think understood) all of yours...

Thanks

Dick

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For instance, if some tells you that:

dN/dt = -λ

then you know that N decreases (decreases because of the sign on lamda) exponentially in time. Notice you didn't need to solve it with calculus, you just needed to remember that differential equations of the form:

dx/dy = K = constant

have exponential solutions

x = Ae^{-Ky
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^{There is a mistake in this portion. I believe that you meant to write}

^{ydx/dy = K has exponential solutions. As you have it written, there would be linear solutions as you pointed out at the beginning of your article.}

^DaMoNarch

You're correct that I was mistaken. I should have written:

dN/dt = -λN

which is the differential equation that has an exponential solution. I have corrected my mistake in my original response. Thank you for catching it.

You wrote:

^{ydx/dy = K has exponential solutions}

dx/dy = (1/y) k

∫dx = k∫ (1/y) dy

x = k lny

x = k(lny)

Your differential equation has a logarithmic solution.

What does the Gaussian function relate to? All the other graphs are sort of familiar/common functions. I have never seen the Gaussian one before.

I suppose that equation has got some special engineering or mathematical significance?

Mike Kovacik,

Joburg South Africa

(2d/3d draughtsman & permanent engineering student!)

Mike,

One of its most common usages is in probability/statistics where it represents the density function of the "normal distribution": it defines the bell shaped curve.

For more information and applications:

http://en.wikipedia.org/wiki/Gaussian_function

Greg

Greg

Thanks for that quick answer. I thought it looked vaguely familiar. I have just found it in one of my engineering mathematics books. Thanks for that website address.

Mike

Hello Mike,

My answer to your question mat not be very popular as it has to do more with the philosophical aspects, rather than the mathematical. As a common answer it could be said that that the Gaussian function represent the NATURAL DISTRIBUTIN of things. Take for example a mathematical description of a weather storm, from very low values through an apex and back to the starting low values. Or as another, the political distribution of the human society: the extremists Will be represented in both sides of the of the curve, left and right, (and BTW, this is actually where these names are derived from), while the center will represent the majority. The Axes will represent the quantitative values such as number of persons, age distribution belief in world peace, or what have you... Another example could be your own history: age vs. your physical condition. or your career. Start at zero,go through the apex and ends at zero. and so on. Practically speaking this function describes most of the day to day events, chemical reactions, statistics and much more.

The rest of thinking about the Gaussian function I'll leave up to you...

Wangito

p.s Gauss is spelt with double "S"

It goes even beyond that. If you take an infinite number of random non-gaussian distributions and add them together, you get a gaussian distribution!