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!

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
