Defining functions in terms of differential equations

[Sigma057]

I have set myself the task of teaching my Freshman in high school brother Calculus, and today while reviewing some topics I saw something I didn't see before.

To start out, I let y = ln[x] => x = e^y
Obviously, we know that y' = 1/x = e^-y

So, I "discovered" that one can define the natural logarithm as the solution to the IVP

y'[x] = e^-y[x], y[1]=0

I know this is a somewhat trivial example,
But my question is, is there anything to be gained by defining functions in this way in general?

 

[krome]

Absolutely! Check out https://www.physicsforums.com/showthread.php?t=105050. It's kind of a long thread about the proof of the orthogonality of Legendre polynomials. I attached a pdf there (mine is the fifth message) showing such a proof which relies solely on the fact that Legendre polynomials satisfy a particular differential equation.

 

[JJacquelin]

Hi !
almost all elementary and special functions can be defined in several ways, for example : integral, differential equation, series, inverse of an already defined function, etc.
This allows to prove many relationships. Using a convenient definition often simplifies the calculus or developments.
Comming back to the differential équations, a list of the most common cases is presented in the pdf "Safari in the Contry of Special Functions":
http://www.scribd.com/JJacquelin/documents
The differential equations appear pp.30-36 in Table 1 : usual functions, Table 2 : special polynomials, Tables 6-9 : examples of special functions.

 

[HallsofIvy]

You can, for example, define "cos(x)" as "The solution to the differential equation, y''= -y, with initial conditions y(0)= 1, y'(0)= 0". You can similarly define "sin(x)" as "The solution to the differential equation, y''= -y, with initial conditions y(0)= 0, y'(0)= 1.

You can, then, prove all of the properties of those functions from those definitions.

This is better, for example, than just "arbitrarily" extending the usual "right triangle" definitions, which, of course, require that the angle be between 0 and 90 degrees, to any value of x.

 

[CellCoree]

Defining functions in terms of differential equations is a common approach in mathematics, particularly in the field of calculus. It allows us to understand the behavior of functions in a more precise and rigorous way. In your example, you have shown that the natural logarithm can be defined as the solution to a specific initial value problem. This approach can be useful in understanding the properties of the natural logarithm and how it relates to other functions.

In general, defining functions in terms of differential equations can provide a deeper understanding of their behavior and relationships to other functions. It also allows us to solve problems and make predictions using mathematical models. This approach is widely used in various fields, such as physics, engineering, and economics, to describe and analyze complex systems.

Furthermore, defining functions in terms of differential equations can also lead to the development of new mathematical techniques and concepts. For example, the concept of a derivative was originally developed to solve differential equations, but it is now an essential tool in calculus and other areas of mathematics.

Overall, while it may seem like a trivial example, defining functions in terms of differential equations can provide valuable insights and applications in mathematics and other fields. It is a powerful tool that allows us to better understand and describe the world around us. As you continue to teach your brother calculus, I encourage you to explore this concept further and see how it can be applied to other functions and problems.