Understanding Hyperbolic Functions from Scratch

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 practically utilised in a wide range of fields. Here are some representative examples:

- Catenary Curves

The shape of flexible objects that hang freely, such as bridges and power lines, is actually represented by the hyperbolic cosine function cosh x. For example, the shape traced by the cables of a suspension bridge is a typical catenary curve, and structural calculations are performed by analysing this shape.

- Vibration and Wave Phenomena

Some physical phenomena are modelled as vibrations or decay involving exponential time variations, and the solutions to these often involve hyperbolic functions. For example, in certain damped vibrational systems, solutions are expressed in terms of combinations of e^x and e^{-x}, and organising these into hyperbolic function form simplifies the analysis.

- Solving Differential Equations

Hyperbolic functions serve as an exemplary role in solving the basic differential equation y'' - y = 0. This allows students to learn the relationship between exponential functions, linear algebra, and eigenvalue problems, fostering a foundation in applied mathematics.

- Growth Models

When approximating phenomena involving rapid growth or decline over a short period, hyperbolic functions can be used. For example, when the utilisation of a resource or the population dynamics of a species changes exponentially in its initial stages, representing the model solution in the form of a hyperbolic function provides a method for deepening understanding of non-linearity.

3. What are the Benefits of Learning?

There are many benefits to learning about hyperbolic function differential equations and their applications, including:

- Strengthening Mathematical Fundamentals

Hyperbolic functions are closely related to exponential functions, and learning about their derivatives and integrals solidifies a foundational understanding of calculus. This further leads to an understanding of the general solutions to linear differential equations.

- Improving Problem-Solving Skills

By learning the process of solving physical phenomena and engineering vibrations using hyperbolic functions when modelling actual problems, your approach to complex phenomena and analytical skills will improve.

- Acquiring Interdisciplinary Skills

Hyperbolic functions are used in many fields, including physics, engineering, economics, and applied mathematics. Therefore, learning their theory and usage leads to cross-disciplinary skill enhancement. This will make model building and simulation on the ground smoother.

- Developing Deep Mathematical Insight

Through hyperbolic functions, you can intuitively understand the relationship with exponential functions, the solution methods for differential equations, and the differences between symmetry and periodicity in mathematics and “aperiodic” changes. This fosters logical thinking and advanced mathematical insight.

In Summary

Hyperbolic function differential equations are expressed in the simple form y'' - y = 0, with solutions such as cosh x and sinh x appearing.

By thoroughly learning about hyperbolic function differential equations, not only will your theoretical understanding deepen, but your analytical ability for actual phenomena will also improve, and your mathematical and engineering skills will be further enhanced.

If you want to learn about the "differential equations" of hyperbolic functions, we recommend this book (access here).