yasir_ali86,
A complete answer to your question would be rather lengthy, but here are some examples of situations where differential equations are useful or necessary:
1
The fundamental relationship between current and voltage in a capacitor involves a first order derivative: I = Cdv/dt.
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The fundamental relationship between current and voltage in an inductor involves a first order derivative: V = Ldi/dt.
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The solution to the response of many filter circuits involves a 2nd order or higher differential equation.
2
Example: You want to find the response of a simple RC lag filter to a step input. The voltage vs time curve depends on whether the voltage to which the capacitor was initially charged affects the output.
3
You need the response to an electrical circuit (a filter, for example) that is described by a differential equation. The use of Laplace transforms reduces the solution to a simple algebra problem. Furthermore, initial conditions are included in the LaPlace transform of the system and its input, so you don't have to handle them separately.
The book "Laplace Transforms aand Control Systems for Technology has answers to all your questions. I believe that it is currently out of print, but is available used for a reasonable price. Amazon has one for sale for $0.46 (yes, that's 46 cents US)
Regards,
Kral