differential equations
Well, your sudden question sounds very strange for me. But don't worry, I'll help you with pleasure.
First of all, the necessity of solving ordinary differential equations arises in various branches of theoretical mathematics, such as: numerical analysis, evaluation mathematics, mathematical analysis, theory of complex variables functions, etc. Unless we possess the techniques of solving these equations, we won't able to cope with other actual mathematical problems.
Besides, in chain theory every linear system may be represented and described by ordinary differential equation. Knowing them, we can analyze the chain in time domain, define yhe output signal for arbitrary input one (Duamel integral), etc.
Also differential equations are widely used in chemistry, mechanics, physics, signal processing, etc.
My sincere advice - take any fundamental book, which covers differential equations. Perhaps, all of yopur doubts and misunderstandings will disappear
With respect,
Dmitrij