Introduction
Our era is one of incredible technological growth. While physicists can
explain the behavior of waves, pendulums, or chaotic systems, engineers
can build robots, and we can communicate wirelessly over a massive global
network. However, the secret to these contemporary marvels is a profound
and enigmatic force called differential equations. However, what exactly are
differential equations? Differential equations, or equations involving
derivatives of a function or functions, are expressions that involve one or
more terms using derivatives of one variable (dependent variable) with
respect to another variable (independent variable). They are remarkably
good at predicting the environment we live in. They can be used to explain a
wide range of phenomena, including bank interest, species population
expansion, changes in investment returns over time, radioactive decay,
continuous compound interest, flow, cooling, and heating issues, and much
more. We will now discuss the applications of first order differential
equations, which fall under the category of differential equations. Let's start
with an explanation of what a first order differential equation is.
An equation of two variables, x and y, with its function f ¿ ) specified on
a region in the xy-plane, is called a first-order differential equation. It is
expressed as dy /dx =f (x , y) . Since there are no higher-order derivatives and
only the first derivative, dy/dx, the equation is of the first order. The
differential equation in first-order can also be represented as; y ’=f ¿) or
(d /dx ) y=f (x , y ). Its applications include, Quantities Varying Exponentially
which is about the general idea of instead of solving equations to find
unknown numbers, we solve equations to find unknown functions. The
second one is Decomposition and Growth, in which are significant
mathematical tools for understanding dynamic processes characterized by
expansion and decay. The third one is Newton’s Law of Cooling, in which it
describes the rate at which an exposed body changes temperature through
radiation. The fourth one is Mixing (Non-Reacting Fluids), they’re word
problems that require us to create a separable differential equation based
on the concentration of a substance in a tank. The last one is Electric
Circuits in which its first-order differential equations arise from the
fundamental relationships between voltage, current, and energy storage
elements such as resistors, capacitors, and inductors.
It's understandable why some people found it challenging given their
extensive and sophisticated applications. But despite their extraordinary
capacity, humans occupy their time and energy deciphering and
characterizing their actions. Some people dedicate their youth to studying
it, grasping its ideas, and delving further into its intricacies. Understanding
its traits and activities, which have a significant impact on our
surroundings, is both worthwhile and difficult. In this section, we will
examine the applications of first order differential equations and learn
about it.
Mathematical Thoughts. (2022, March 7). The importance of differential equations.
https://mathematicalthoughtsdot.wordpress.com/2018/06/30/the-importance-of-
differential-equations/
BYJU’S. (2020, August 27). First Order differential equation (solutions, types & examples).
BYJUS. https://byjus.com/maths/first-order-differential-equation/#applications
