Solution to Challenge Problem: f'(x) = f(x) for all x in Q

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In summary, the conversation discusses a challenge problem posted on a Harvard Math 25a webpage where the question asks if a function f(x) satisfying f'(x) = f(x) for all rational x must be the zero function. The participants also discuss other interesting challenge problems on the webpage and verify that the question is not from an assignment. The solution to the problem is that the only function from Q to Q satisfying the differential equation is f(x) = 0.
  • #1
altcmdesc
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This isn't a homework question or anything, but I came across this challenge problem posted on a Harvard Math 25a webpage and I'm wondering what the solution to it is since no solution is posted on the page.

Suppose that [tex]f\colon \mathbb{Q} \to \mathbb{Q}[/tex] satisfies [tex]f'(x) = f(x)[/tex] for all [tex]x \in \mathbb{Q}[/tex]. Must [tex]f[/tex] be the zero function?
 
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  • #2
altcmdesc said:
This isn't a homework question or anything, but I came across this challenge problem posted on a Harvard Math 25a webpage and I'm wondering what the solution to it is since no solution is posted on the page.

Suppose that [tex]f\colon \mathbb{Q} \to \mathbb{Q}[/tex] satisfies [tex]f'(x) = f(x)[/tex] for all [tex]x \in \mathbb{Q}[/tex]. Must [tex]f[/tex] be the zero function?

Could you please provide a link to where this came from? For it to stay in the general technical forums, we would need to verify that it is not from an assignment.
 
  • #3
Sure:

http://www.math.harvard.edu/~tomc/math25a/challenge.pdf

It's from a class held Spring 2005.

There are other challenge problems posted on the course webpage which are interesting as well.
 
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  • #4
altcmdesc said:
Sure:

http://www.math.harvard.edu/~tomc/math25a/challenge.pdf

It's from a class held Spring 2005.

There are other challenge problems posted on the course webpage which are interesting as well.

Verified, thank you. Good luck!
 
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  • #5
I would say it must be, but hopefully I'm not invoking too strong theorems.

Solving the differential equation provided, the family class is [tex]f(x) = ae^x[/tex] where a is some real constant. However for any rational argument, the exponential function has an irrational (in fact, transcendental?) value. So the only function from Q to Q will be the case where a=0, ie f(x) =0.

EDIT: The first problem is also relatively straightforward, seeing as its given that the series are absolutely convergent, so we can rearrange terms, those results follow in the same way the finite cases do.
 
  • #6
That's the conclusion I reached as well, but I figured that it was too simple of an answer to a challenge problem..
 

FAQ: Solution to Challenge Problem: f'(x) = f(x) for all x in Q

What is a Challenge Problem?

A Challenge Problem is a complex and often open-ended scientific question that requires innovative thinking and problem-solving skills to solve. It is designed to push the boundaries of current scientific knowledge and often involves collaboration between scientists from different disciplines.

How are Challenge Problems chosen?

Challenge Problems are typically chosen by a team of experts in a specific field or by a funding agency based on its relevance, difficulty, and potential impact on society. They may also be proposed by individual scientists or research teams.

What makes a good Challenge Problem?

A good Challenge Problem should be well-defined, relevant, and have the potential to advance scientific knowledge in a significant way. It should also have real-world applications and be feasible to solve within a reasonable time frame.

How do scientists approach solving a Challenge Problem?

Scientists typically approach solving a Challenge Problem by breaking it down into smaller, more manageable sub-problems and conducting research to gain a better understanding of the problem and its underlying mechanisms. They may also collaborate with other scientists and use a variety of techniques and technologies to tackle the problem.

What are some examples of successful Challenge Problems?

Some examples of successful Challenge Problems include the Human Genome Project, which aimed to sequence the entire human genome, and the Higgs Boson discovery, which confirmed the existence of the Higgs Boson particle. Both of these projects required collaboration between scientists from different fields and resulted in significant advancements in our understanding of genetics and particle physics.

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