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I know HOW to mechanically solve basic diff. equations. To recap, you start out with the derivative $\frac{dy}{dx}=...$ and you aim to find out y=... To do this, you separate the variables, and then integrate.

But, can someone give me a some context? A simple example or general sense of WHY you solve a differential equation. Know a common situation they are used to model, and then the purpose of then finding the original function from whence the derivative came? When do you initially know the derivative? When you only know the rate of change?

Thanks!

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    $\begingroup$ Physical systems are often modelled by differential equations. $\endgroup$
    – Daniel Fischer
    Jan 13, 2014 at 13:39
  • $\begingroup$ ... as are economic and biological systems. The Question as currently stated is terribly broad. $\endgroup$
    – hardmath
    Jan 13, 2014 at 13:44
  • $\begingroup$ Kepler laws have been proved and understood by solving equations of motion, which are differential equations of second order. This seems to be one of the earliest problems, for which analysis was invented. $\endgroup$
    – Ruslan
    Jan 13, 2014 at 13:44
  • $\begingroup$ See this list of examples: en.wikipedia.org/wiki/… $\endgroup$
    – Ruslan
    Jan 13, 2014 at 13:47
  • $\begingroup$ One of the easiest examples that I remember from gymnasium: en.wikipedia.org/wiki/Simple_harmonic_motion $\endgroup$
    – Carsten S
    Jan 13, 2014 at 13:50

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Basically almost every phenomenon that is continuously time-dependant and determinist (nothing random) is modelled by a differential equation. There are examples in physics, chemistry, biology, economics, you name it. A few examples among trillions :

  • mecanics : the movement of a point subject to a force F is determined by differential equation $m \ddot x = F(x)$ where $m$ is the mass.

  • electronics : the intensity in a circuit with resistances and capacitors is determined by a differential equation

  • chemistry : the advancement of a chemical reaction is given by a differential equation involving the concentrations of the products

  • thermodynamics : the temperature of your coffee mug left in a constant temperature room is determined by a differential equation

etc.

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  • $\begingroup$ Sorry but although this answer states areas DEs are used, it fails to make one understand what are the DEs for. Same answer could be inserted for a question like why is multiplication/division/whatever used. See @kleineg's answer. $\endgroup$
    – akostadinov
    Jan 13, 2014 at 16:17
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I think this question comes from a disconnect between math in the classroom and math in science/industry.

Say you had data for some variable as it changes in time, if you wanted to predict what the value would be in the future your best bet would be to estimate the derivative and solve the differential equations.

Now, this is how differential equations were used to form basic physical laws, from the existing data. In physics, finance, and social sciences they use equations that have been shown to hold in general circumstances to predict the behavior of variables in other specific circumstances. Ruslan provided a great link to a list of uses for differential equations.

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  • $\begingroup$ I'm following the principle, but nothing illustrates/reinforces better than a concrete example. Can we work through a simple contrived example? How about a situation modelling age vs. salary? Here are some data points: (age,salary in thousands) (20,30) (25,55) (30,225) (35,75) (40,100) (45, 110) (50,120) (55,130) (60,140) What do we do next to get a diffEQ, and then can we predict income at age 65? or 43? $\endgroup$
    – JackOfAll
    Jan 14, 2014 at 14:38
  • $\begingroup$ Also, when I use existing data points to estimate unknown ones, I tend to use scatter plots, correlation, and a regression equation. Not seeing how this translates into DiffEq's. $\endgroup$
    – JackOfAll
    Jan 14, 2014 at 17:09
  • $\begingroup$ @JackOfAll I have been thinking hard since your first comment because the first thing I thought of from your example is regression analysis. $\endgroup$
    – kleineg
    Jan 14, 2014 at 17:50
  • $\begingroup$ The cheap answer would be that using regression involves fitting data to an unknown differential equation, but it is unsatisfactory because you don't use any of the same tools. I would say that a better case would be where I was measuring the rate of change of a variable (like speed/velocity measurements from a speedometer), but in the case of kinematics I already know the solutions to the differential equation and would just plug in the variable. I have calculated numerical solutions in Physics courses, that was more starting with an equation and generating a solution. $\endgroup$
    – kleineg
    Jan 14, 2014 at 18:00
  • $\begingroup$ My background was as a double major in Math and Physics who is working as a programmer for engineers. The engineers do use differential equations coming from Physics to get algebraic equations that the computer actually performs. For example heat flux is the total heat flow now, minus the heat flow a second ago, divided by the second. And then that heat flux is put into another equation. The equations are decomposed before being programmed, and the solutions are already known. $\endgroup$
    – kleineg
    Jan 14, 2014 at 18:11
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There are very much laws in physics that use differential equations. For example, Newton's second law can be written as: $$F=\dfrac{dp}{dt}=\dfrac{d{(mv)}}{dt}$$ Many times the force is a function of the location (as in spring). The location is the derivative of the velocity, so the movement gives a differential equation.

For more complete list, see that wiki article

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In the introduction of Arnold's book, talking about differential equations:

[...] Newton considered this invention of his so important that he encoded it as an anagram whose meaning in modern terms can be freely translated as follows: “The laws of Nature are expressed by differential equations.”

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ok let us consider following situation,in signal processing for continuous systems,input and output is related by ordinary differential equation,while in discrete case it is difference equation,for example lapalce trasnform is used to convert differential equation into algebraic equation,also fourier transform is used to solve differential equation

http://books.google.ge/books?id=_xeQNRlxzG4C&pg=PA105&lpg=PA105&dq=differential+equation+in+signal+processing&source=bl&ots=1dJWhR4wwQ&sig=waYGonRZfEITmLzVMh7H0ZiMgwg&hl=ka&sa=X&ei=x_DTUsyjOMWGtAabm4DwBA&ved=0CFQQ6AEwBQ#v=onepage&q=differential%20equation%20in%20signal%20processing&f=false

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