Why - not how - do you solve Differential Equations? 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!
 A: 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.
A: 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.
A: 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
A: 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.”

A: 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
