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Here’s an in-depth blog post exploring linear differential equations with constant coefficients, from basic concepts and applications to the benefits of learning them. We hope this will demonstrate the importance of this field to anyone interested in mathematics, engineering, or physics. 1. What are Linear Differential Equations with Constant Coefficients? Linear differential equations with constant coefficients are differential equations where the unknown function and its derivatives appear linearly, and the coefficients do not depend on the variable but are constant values.  The general solution to these equations is found by assuming a form of *y* = e^*rx* and solving the corresponding characteristic equation. This method allows even complex differential equations to be solved analytically. 2. Where are They Used? Due to their ease of handling and consistent solution methods, linear differential equations with constant coefficients are applied to a wide ran...

1. What are Differential Equations Solved with Variation of Parameters? Differential equations are a powerful tool for mathematically capturing changing phenomena. Among these, the “method of variation of parameters” is a particularly useful technique for solving non-homogeneous linear differential equations. The general solution to a homogeneous differential equation is known, expressed by a combination of constants (constant coefficients).  However, this cannot be directly solved when a non-homogeneous term (corresponding to an external influence or input) is added. Therefore, the method of variation of parameters takes an approach of replacing the original constant parts with (unknown) functions and determining the shape of those functions through differentiation. This method allows the construction of a complete solution including the non-homogeneous term.  Due to its flexibility in handling various systems – such as when the non-homogeneous term is an exponential function...

In the fields of mathematics and engineering, differential equations are a powerful language for capturing the “motion” and “change” of natural and social phenomena. Among these, hyperbolic functions (such as cosh, sinh, tanh, etc.) are defined using exponential functions and possess unique properties. Notably, these hyperbolic functions frequently appear as general solutions to simple second-order linear differential equations, making them easy to analyse and having a vast number of applications. This article will first explain what “hyperbolic function differential equations” are, then explore the situations in which they are used, and finally outline the benefits of learning them. 1. What are Hyperbolic Function Differential Equations? Hyperbolic functions are combinations of exponential functions. They are positioned as important specific examples of certain differential equations. 2. Where are They Used? Hyperbolic functions and their associated differential equations are practica...

Differential equations are a vitally important tool for modelling ‘change’ which appears in a wide range of fields, including natural sciences, engineering, economics, and life sciences. Amongst these, differential equations of separable variables represent one of the simplest classes where a straightforward solution method can be applied. Here, we will look in detail at what a differential equation of separable variables is, the specific scenarios in which it is used, and the benefits of studying this field. 1. What are Differential Equations of Separable Variables? A differential equation of separable variables is one where, within a given differential equation, the variables *x* and *y* (or other variables) can be separated into one side of the equation.  If the condition that “the variables can be successfully separated” is met, the problem can be solved relatively simply and directly, making it popular with beginners. 2. In What Scenarios are They Used? Differential equations ...

The reciprocal function is one of the fundamental functions in mathematics, and despite its simplicity, it’s a powerful tool with applications in many fields thanks to its unique characteristics. This article will provide a detailed explanation of the definition and properties of reciprocal functions, explore the contexts in which they are used, and outline the benefits of learning about them. 1. What is a Reciprocal Function? A reciprocal function returns the reciprocal of a given real number.  - Graph Shape The graph of a reciprocal function forms a hyperbola, with values increasing or decreasing rapidly as it approaches the origin. It takes the shape of a hyperbola spanning the first and third quadrants, and has asymptotes at x = 0 and y = 0. Behind this simple equation lies the concept of a multiplicative inverse, which forms the foundation of basic algebra. 2. Where are Reciprocal Functions Used? Due to their fundamental nature and simplicity, reciprocal functions are used in ...

Logarithmic functions are a powerful mathematical tool, acting as the inverse of exponential functions, allowing us to replace multiplicative relationships with additive ones and understand the fluctuations in the magnitude of numbers. When dealing with the rapid growth or decay of exponential functions, logarithmic functions play a role in compressing the scale of data and making complex phenomena easier to understand. This article explains what logarithmic functions are, where they are used, and the benefits of learning them. 1. What are Logarithmic Functions? A logarithmic function indicates the inverse operation of a positive real number raised to a power. - Inverse of Exponential Functions Logarithmic functions perform the inverse operation of exponential functions to derive values. - Scale Transformation Taking the logarithm transforms multiplicative relationships into additive ones. This makes it easier to handle very large values or rapid changes. 2. Where are they Used? Logari...

Exponential functions are a remarkably powerful tool for understanding mathematical models and real-world phenomena. They can vividly express phenomena we see in our daily lives, science, and the world of economics, such as “rapid growth” or “sharp decline,” in a simple yet effective way. Here, we will delve deeply into what exponential functions are, where they are used, and the many benefits of learning them. 1. What are Exponential Functions? Exponential functions are functions where the variable appears in the exponent. - Expressing Rapid Growth and Decay Exponential functions are suitable for modelling phenomena where values change exponentially over time, such as population growth, compound interest, and radioactive decay. - Continuous and Smooth Curves The graph of an exponential function changes continuously and smoothly, making it highly suitable for analysing limits, derivatives, and integrals. This allows dynamic changes in physical or economic phenomena to be analysed using...

Trigonometry is a particularly deep and widely applicable field within mathematics. Originating in ancient Greek astronomy and surveying, it is now an indispensable tool in fields ranging from modern engineering and physics to information technology. This article will first explain the basic concepts of “What is trigonometry?”, then look at how it is used in various situations, and finally explain the benefits of learning trigonometry. 1. What is Trigonometry? Trigonometry is a set of functions that express the relationship between angles and side lengths in a right-angled triangle. The most well-known of these are sine (sin), cosine (cos), and tangent (tan). - Definition in a Right-Angled Triangle In a right-angled triangle, trigonometric functions are defined by the ratios of the sides opposite, adjacent, and hypotenuse to a given acute angle. For example, for an angle θ:   - sin θ = Opposite ÷ Hypotenuse   - cos θ = Adjacent ÷ Hypotenuse   - tan θ = Opposite ÷ Adjacent...