Differential Equations: An Introduction

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Differential Equations hold a very special place in the field of mathematics, forming the strong interlocking chain that bridges the gap between abstract mathematics and each of the practical sciences. Principally, differential equations are a subset of calculus, as every differential equation works with functions and their derivatives. Much like algebraic equations, some differential equations are easy enough to be solved by any novice, while others have remain unsolved and teased centuries of geniuses.


WHAT IS A DIFFERENTIAL EQUATION?

Differential equations are any equations that connect a function and its derivatives (of any order) together. Naturally, the most basic quality of a differential equation is that there are no numerical answers to it. Rather, a differential equation always has solutions in the form of a function/a set of functions. 

                       

As seen above, there are various ways to represent the same differential equation, owing to the different ways of denoting a derivative. Solving differential equations can sometimes be a nightmare, and hence the first step is always to examine the complexity of the equation based on its order.


WHAT IS THE ORDER OF A DIFFERENTIAL EQUATION?

The order of a differential equation is always the highest order of the derivative of any function in a certain equation. Hence, first locate the highest derivative and then find its corresponding order. In the equation below, the order would be 3.

                            


Differential equations hold an important place in each of the natural sciences, as mentioned earlier. In physics, almost every theory is backed up by differential equation, apart from the fact that kinematics (study of motion) is purely differential equations. In chemistry, the rate of every reaction is modelled using a set of differential equations and even in biology, cellular decay and cancer growth is monitored with differential equations. Learn about differential equations in its entirety through this oversimplified presentation.

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