Differential equations are mathematical equations that describe how a variable changes over time. They can be used to model a wide range of phenomena in the real world, such as the spread of diseases, the movement of celestial bodies, and the flow of fluids.
One of the key features of differential equations is that they can account for the many factors that can influence the variable being studied. In the field of engineering, differential equations are commonly used to design and analyze systems such as electrical circuits, mechanical systems, and control systems. In the natural sciences, differential equations are used to model the evolution of physical systems over time.
Overall, differential equations play a vital role in our understanding of the world around us, and they are a powerful tool for predicting and controlling the behavior of complex systems. Summarized below are some crucial and common applications of the differential equation from real-life.
Applications of differential equations
Mathematics has grown increasingly lengthy hands in every core aspect. Differential equations have aided the development of several fields of study.
Hence, just like quadratic equations, even differential equations have a multitude of real-world applications. Some of the most common and practical uses are discussed below.
1. Bonds between atoms and molecules
Everything we touch, use, and see comprises atoms and molecules. In actuality, the atoms and molecules form chemical connections within themselves that aid in maintaining their cohesiveness.
Atoms are held together by chemical bonds to form compounds and molecules. Covalent, polar covalent, and ionic connections are all types of chemical bonding. The second-order differential equations are used to express them. This equation comes in handy to distinguish between the adhesion of atoms and molecules.
2. Systems of the electric circuit
The absolute necessity is lighted in the dark and fans in the heat, along with some entertainment options like television and a cellphone charger, to mention a few. But how do they function? Electric circuits are used to supply electricity.
The differential equation of the same type determines a circuit consisting of an inductance L or capacitor C and resistor R with current and voltage variables. The differential equation is regarded as conventional when its second order, reflects the derivatives involved and is equal to the number of energy-storing components used.
3. Population growth
Recording the population growth rate is necessary since populations are growing worldwide daily. We thus take into account the most straightforward differential equations model available to control a particular species’ population dynamics.
The equation will give the population at any future period. Even though it does not consider numerous variables like immigration and emigration, which can cause human populations to increase or decrease, it proved to be a very reliable population predictor.
4. To describe the motion of pendulums
Every home has wall clocks that continuously display the time. Have you ever observed a pendulum that swings back and forth constantly without pausing? If so, how would you characterize the motion?
Newton’s second law of motion is used to describe the motion of the pendulum from which a differential equation of second order is obtained. Forces acting on the pendulum include the weight (mg) acting vertically downward and the Tension (T) in the string.
T = mg cos θ
When a pendulum is displaced sideways from its equilibrium position, there is a restoring force due to gravity that causes it to accelerate back to its equilibrium position. This restoring force causes an oscillatory motion in the pendulum.
Restoring force = m.d2x/dt2 = -mg sin θ
5. Exponential growth of bacteria
Microorganisms known as bacteria are so tiny in size that they can only be observed under a microscope. They are present in the air, soil, and water. Surprisingly, they are even present in large numbers in the human body. Some make us healthy, while others make us sick.
As is often said, nothing in excess is inherently desirable, and the same is true with bacteria. But differential equations assist us similarly when trying to detect bacterial growth.
6. To find the Velocity and Acceleration
Differential equations can be used to describe the relationship between velocity and acceleration, as well as other physical quantities. For example, Newton’s second law of motion states that the acceleration of an object is directly proportional to the force acting on it and inversely proportional to its mass. This relationship can be written as a differential equation in the form:
F = ma
where F is the force acting on the object, m is its mass, and a is its acceleration. By solving this differential equation, we can determine the acceleration of an object as a function of time, given the forces acting on it and its mass.
Similarly, we can use differential equations to describe the relationship between velocity and acceleration. For example, the relationship between velocity and acceleration can be described by the equation:
a = dv/dt
Which forms the above equation as
F = m.dv/dt
where a is the acceleration, v is the velocity, and t is time. By solving this differential equation, we can determine the velocity of an object as a function of time, given its acceleration.
In general, differential equations are a powerful tool for describing and analyzing the behavior of physical systems that change over time, and they are widely used in a variety of fields, including physics, engineering, and economics.
7. To determine radioactivity decay
Differential equations can be used to describe the rate of decay of radioactive isotopes. Radioactive decay is a random process, but the overall rate of decay for a large number of atoms is predictable. The rate of decay for a particular isotope can be described by the differential equation:
dN/dt = -λN
where N is the number of atoms of the isotope at time t, and λ is the decay constant, which is characteristic of the particular isotope. The negative sign in this equation indicates that the number of atoms decreases with time as the isotope decays.
By solving this differential equation, we can determine the number of atoms of the isotope remaining at any time t, given the initial number of atoms and the decay constant. This is useful for predicting the behavior of radioactive isotopes and understanding their role in various applications, such as medicine and power generation.
Strategies to teach differential equations effectively
The task for the lecturer is to create a link between abstract mathematical ideas and real-world applications of the theory. Finding the ideal balance between a grasp of mathematics and its applications in one’s particular subject is essential for successfully teaching a particular concept.
Since many real-world applications employ differential equations as mathematical models, a course on ordinary differential equations works rather well to put this constructing the bridge idea into practice.
Innovative strategies are needed to raise student engagement and performance in mathematics classrooms. Various strategies that have proved to be effective are as follows:
1. Technology
Technology can be used in various ways, depending on institutional restrictions, available resources, and instructor preferences, such as a teacher-led demonstration tool, a lab activity carried out outside of class time, or an integrated component of regular class sessions.
2. Modelling Process
Students must translate an issue from a real-world situation into a mathematical model, solve that model, and then apply the solutions to the original problem. Students are asked to create the equation or the model’s heuristics rather than being given the model or algorithm and instructed to enter numbers into the equation to discover the solution. To create a model, it is crucial to define variables with the correct units, state what is known, make reliable assumptions, and identify the problem at hand.
3. Student & feedback
Positive student feedback has been helpful in encouraging students. When students can use their math skills to solve issues they could see again in a scientific or engineering course, they are more likely to acquire the material. According to course-ending polls, students undergo a metamorphosis once they perceive that the lectures and evaluations are focused on issues they could face in the real world. Students believe that the lessons are more engaging.
Additionally, they think that when they apply mathematics to real-world issues, their confidence levels increase because they can feel if the solution makes sense. They realize that reasoning abilities are just as crucial as analytical abilities. They can get some credit for describing what their intuition tells them should be the solution if they are sure in their model and get an answer that just does not make sense.
Conclusion
Due in part to growing interest in dynamical systems and a general desire to enhance mathematics learning and instruction, the teaching and learning of differential equations are moving in new directions.
The use of technology, which requires that ideas and approaches be approached graphically, numerically, analytically, and descriptively, modeling, and student feedback is a springboard for considering new techniques for helping students understand the fundamental concepts and approaches in differential equations.
Research into students’ thinking and reasoning is producing fresh insights into establishing and maintaining learning settings where students may develop a profound comprehension of mathematical ideas and procedures, in addition to novel pedagogical tactics.
An engineer, Maths expert, Online Tutor and animal rights activist. In more than 5+ years of my online teaching experience, I closely worked with many students struggling with dyscalculia and dyslexia. With the years passing, I learned that not much effort being put into the awareness of this learning disorder. Students with dyscalculia often misunderstood for having just a simple math fear. This is still an underresearched and understudied subject. I am also the founder of Smartynote -‘The notepad app for dyslexia’,