Family of Curves w/ Slope=1 at (1,1) - Can You Answer?

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Just a question that came to my mind while studying differential equations.

Of-course this is a silly one (I think), but I wonder if someone can answer it!

Thanks :smile:
 
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I'm not sure what you mean by "find" that family. A perfectly good answer is "the family of all f(x) such that f(1)= 1 and f'(1)= 1". But there are simply too many of them to be able to write a "formula".
 

Thread 'Volterra equation of the second kind'

There is the following linear Volterra equation of the second kind $$ y(x)+\int_{0}^{x} K(x-s) y(s)\,{\rm d}s = 1 $$ with kernel $$ K(x-s) = 1 - 4 \sum_{n=1}^{\infty} \dfrac{1}{\lambda_n^2} e^{-\beta \lambda_n^2 (x-s)} $$ where $y(0)=1$, $\beta>0$ and $\lambda_n$ is the $n$-th positive root of the equation $J_0(x)=0$ (here $n$ is a natural number that numbers these positive roots in the order of increasing their values), $J_0(x)$ is the Bessel function of the first kind of zero order. I...
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