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!

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.”

