Any insights on this equation?

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In summary, the conversation is about an equation with the form f(t')|_0^t = \int_0^t f(t')dt' and the interpretation of such an expression. The participants discuss the characteristics of f(t) that satisfy this equation and whether it needs to hold for all values of t or only for some. They also consider different functions that could potentially satisfy the equation and any additional constraints. Ultimately, they are trying to determine under what conditions there is a solution or no solution to the equation.
  • #1
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Hi. Sorry I couldn't think of a more appropriate title for this thread.

I'm doing some calculatons and I arrived at an equation with this form
[tex]
f(t')|_0^t = \int_0^t f(t')dt'
[/tex]

I was just wondering if anyone as any insight into the interpretation of such an expression. All I can think of is the obvious: that the difference between the function f(t) at the endpoints must equal the area under the curve f(t) between those endpoints. I guess what I'm asking is, what characteristics must f(t) have in order to satisfy this.

Moreover, and probably importantly, I only need this to hold from SOME value of t, not ALL values of t.

Thanks.
 
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  • #2
Well it looks to me like f(t') = f'(t')
 
  • #3
What function is it's own antiderivative? Maybe do without the prime in there cus' that's just confusing. Write it as:

[tex]f(x)\biggr|_0^t=\int_0^t f(u)du[/tex]
 
  • #4
Thank you for your replies.

The function f(x) = e^x would do the trick, but then the equation is satisfied for ALL t. I'm particularly interested in cases where the equation is satisfied for discrete values of t.

Think of it like this: The left hand side and right hand side are DIFFERENT functions (e.g. g(t) and h(t)). Under what conditions would these functions intersect at some value of t?
 
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  • #5
I don't think I know how to help you, sorry.
 
  • #6
No problem. You don't have to apologize. I just thought that maybe someone would be all like "yeah, that means...blah blah...compact support...blah blah...and the supremum norm of f^-1 would have to be...blah blah...least upper bound...blah blah with transendality four... blah blah" and all that other stuff I didn't learn by never taking a proper real analysis course.
 
  • #7
If the equation need be true only for some particular value of t, then most any function will serve. For example, if f(t) = 1, then t = 0 works; if f(t) = t, then t = 2 works; if f(t) = t^2, then t = 3 works; if f(t) = cos(t), then t = π/4 works; etc. Does the problem have any other constraints that weren't mentioned?
 
  • #8
Hmmm. Interesting. I was finding that most "input functions" I tried gave solutions. But I can't seem to find some general condition telling us when there is a solution or not. The variable t represents time, and in this case it should be positive. But other than that there are no additional restrictions. What about if the original equation were replace by
[tex]
e^{x}f(x)|_0^t = \int_0^t e^uf(u)du
[/tex]
and we reverse the question: under what conditions is there NO solution?
 

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