Research article · 2018 · Full text
International Journal of Statistics and Applied MathematicsAn introduction to differential equations
Volume 3(1) · Pages 115–117
Research areasNumerical methods for differential equations
Abstract
Author-writtenWhole of the physics and engineering depends upon differential equations, their laws dominate electronics, mechanical and civil engineering. Independent use of differential equations makes it perfect tool for the use of applied mathematics. Here we are discussing some aspects of the differential equations their definitions, types of solutions, their applications and uses.
Summary
Source-checkedPlain-language reading of the full paper, prepared with AI and checked against the text. Each point links to the authors' own words.
Why it matters
- Differential equations are fundamental to physics and engineering, particularly in the fields of electronics, mechanical engineering, and civil engineering. View evidence ↗
- Because they can be used independently, differential equations serve as an ideal tool for applied mathematics. View evidence ↗
The question
- The paper aims to discuss the definitions, types of solutions, and the various applications and uses of differential equations. View evidence ↗
What they did
- A differential equation is defined as an equation that includes independent variables, dependent variables, and the derivative of the dependent variable with respect to the independent variables. View evidence ↗
- Differential equations are categorized into two main types: Ordinary Differential Equations (ODE) and Partial Differential Equations (PDE). View evidence ↗
- Ordinary differential equations are those that involve derivatives with respect to only one independent variable. View evidence ↗
- Partial differential equations involve partial derivatives with respect to more than one independent variable. View evidence ↗
What they found
- The order of a differential equation is determined by the highest order derivative present in the equation. View evidence ↗
- A differential equation is linear if all dependent variables and derivatives appear only to the first power and no products of dependent variables or their derivatives exist. View evidence ↗
- A solution (or integral) of a differential equation is a relationship between variables that, when substituted into the equation, transforms it into an identity. View evidence ↗
- A general solution of an nth order ordinary differential equation is one that contains n independent arbitrary constants. View evidence ↗
- A singular solution is derived from a general solution by assigning specific values to one or more of the arbitrary constants. View evidence ↗
- In electrodynamics, Maxwell's equations (a set of PDEs) combined with the Lorentz force law provide the foundation for electric circuits, classical optics, and classical electrodynamics. View evidence ↗
- Schrödinger's equation is a linear PDE used in quantum mechanics to describe how a system's wave function evolves over time. View evidence ↗
- The Lotka-Volterra (predator-prey) equations consist of two first-order non-linear differential equations used to model the population dynamics of interacting predator and prey species. View evidence ↗
What it means
- The authors conclude that differential equations are vital for modeling almost every technical, biological, or physical process, ranging from neuron interactions to bridge design. View evidence ↗
- The authors state that when identical differential equations arise from different scientific fields, the underlying mathematical theory acts as a unifying principle for those diverse phenomena. View evidence ↗
Visual summary
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Study overview
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View source evidence ↗Introduction Differential Equation
An equation involving independent, dependent variable and derivative of dependent variable with respect to independent variables is called differential equation. It can be further divided into two categories: 1. O.D.E.
P.D.E
P.D.E
ODE Ordinary Differential Equation
A differential equation involving derivative with respect to a single independent variable is called an ordinary differential equation. for example
PDE Partial Differential Equation
A differential equation involving partial derivatives with respect to more than one independent variables is called a partial differential equation.
For example
)+5cosx=0 is partial differential equation. Here we are providing some basic definitions of ingredients used in differential equations:-
Order of a differential equation
The order of highest order derivative involved in a differential equation is called the order of the differential equation. for example order of dy =(x+sinx)dx is 1
Degree of a differential equation
The degree of a differential equation is the degree of highest derivative which occurs in it, after the dependent has been made free from radicals and fractions as far as the derivatives are concerned .for example degree of
) 5 = 𝑒 𝑡 is 1 as it is the degree of highest order derivative occurring in the equation.
Linear and nonlinear differential equations
A differential equation is said to be linear if:-1. Every dependent variable and every derivative involved occur in the first derivative only. 2. No products of dependent variables and/or derivative occur.
For example dy = (x + sinx) dx is the linear differential equation.
A differential equation which is not linear is called nonlinear differential equation. For example
) 5 = 𝑒 𝑡 is non linear differential equation.
Solution of a differential equation
Any relation between the dependent variables and independent variables, when substituted in the differential equation, reduces it to an identity is called a solution or integral of the differential equation it should be noted that a solution of a differential equation does not involves the derivatives of the dependent variable.
For example y=c𝑒 2𝑥 is the solution of
Particular solution and singular solution
be the nth order ordinary differential equation.
General Solution
A solution of (1) containing n independents arbitrary constants is called a general solution of the differential equation.
Singular Solution: A solution of (1) obtained from a general solution of (1) by giving particular values to one or more of the n independent arbitrary constants is called the singular solution of (1).
Classical mechanics
All we know that, Newton's second law is sufficient to describe the motion of a particle. Once independent relations for each force acting on a particle are available, they can be substituted into Newton's second law to obtain an ordinary differential equation, and various other relations which are called the equation of motion.
Electrodynamics
Maxwell's equations forms a set of partial differential equations that, along with the Lorentz force law, form the base of classical electrodynamics, classical optics, and electric circuits. These fields are the base for modern electrical and communications technologies. Maxwell's equations describe how electric and magnetic fields are generated and altered by each other and by charges and currents. They are named after the Scottish physicist and mathematician James Clerk Maxwell, who published an early form of those equations between 1861 and 1862.
General relativity Quantum mechanics
In quantum mechanics, the analogue of Newton's law is Schrödinger's equation (a partial differential equation) for a quantum system (usually atoms, molecules, and subatomic particles whether free, bound, or localized). It is not a simple algebraic equation, but in general a linear partial differential equation, describing the time-evolution of the system's wave function (also called a "state function").
Biology Verhulst equationbiological population growth von Bertalanffy modelbiological individual growth Replicator dynamicsfound in theoretical biology Hodgkin-Huxley modelneural action potentials
Biology Verhulst equationbiological population growth von Bertalanffy modelbiological individual growth Replicator dynamicsfound in theoretical biology Hodgkin-Huxley modelneural action potentials
Predator-prey equations
The Lotka-Volterra equations, which are also known as the predator-prey equations, are a pair of first-order, non-linear, differential equations frequently used to describe the population dynamics of two species that interact, one as a predator and the other as prey.
Chemistry
The rate law or rate equation for a chemical reaction is a differential equation that links the reaction rate with concentrations or pressures of reactants and constant parameters (normally rate coefficients and partial reaction orders). To determine the rate equation for a particular system one combines the reaction rate with a mass balance for the system. The concept of rate of reaction can be applied to the tumour clots in which we can minimize the rate of growth of tumour with respect to time and we can have more time to cure that by proper medicine. Schrodinger equation could not be possible without the use of differential equation for the concept of electron density function. Partial derivatives of the function ᴪ with respect to the variables x, y and z gives the required equation and are known as Schrodinger equation
Economics
Economics
Conclusion
The study of differential equations is a wide field because of their adaptability in pure and applied mathematics, physics, and engineering. All of these branches are concerned with the properties of differential equations various types.. Differential equations play a vital role in modelling virtually every physical, technical, or biological process, from critical problems, to bridge design, to interactions between neurons. Differential equations such as those used to solve real-life problems may not necessarily be directly solvable, i.e. do not have closed form solutions. Instead, solutions can be approximated using numerical methods and grid generation methods various fundamental laws of physics and chemistry can be obtained from differential equations. In biology and economics, differential equations are used to model the behaviour of complex systems. Theory of differential equations in mathematical form is first developed together with the sciences where the equations had originated and where the results found application. However, diverse problems, sometimes originating in quite separate scientific fields, which give rise to identical differential equations. Whenever this happens, mathematical theory behind the equations can be viewed as a unifying principle behind diverse phenomena. As an example, consider propagation of light and sound in the atmosphere and of waves on the surface of a pond. All of them may be described by the same secondorder partial differential equation, the wave equation, which allows us to think of light and sound as forms of waves, much like familiar waves in the water. Conduction of heat, the theory of which was developed by Joseph Fourier, is governed by another second-order partial differential equation, the heat equation. It turns out that many diffusion processes, while seemingly different, are described by the same equation; the Black-Scholes equation in finance is, for instance, related to the heat equation.
References
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