I am doing a research paper and I faced with this problem. Does anyone know the answer (or an approximation for the answer) for the following integral equation? $$\int_0^{y(x)} e^{xt}f(t)dt = g(x)$$ We want to find $y(x)$. In case it is necessary, we know that $y(0)=0$ and $g(0)=0$. We know that $f(x)$ is a probability density function and $\int_0^\infty$f(x)dx=1.
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3$\begingroup$ Have you tried differentiating with respect to $x$ (if $g$ is regular enough)? $\endgroup$ – Loïc Teyssier Nov 28 '17 at 7:10

$\begingroup$ yes, but it did not end up with a closed form answer. May be I did not do it properly. $\endgroup$ – Mamal Nov 28 '17 at 7:20

4$\begingroup$ Closed form is out of the question in this generality. You're dealing with integration and inverting a function as subproblems, which already don't have explicit answers. Existence of a solution isn't guaranteed either. $\endgroup$ – Christian Remling Nov 28 '17 at 7:38

$\begingroup$ Even a clue of how we can come up with a solution (even if it is not in a closed form) is important for me. $\endgroup$ – Mamal Nov 28 '17 at 7:40

$\begingroup$ @Mamal Define the word "solution". $\endgroup$ – fedja Nov 28 '17 at 17:54