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I've run into having to solve this equation for $f(x)$:

$1+ax=\int_{-\infty}^xf(x-t)dt$

Unfortunately, I am not familiar with solving integral equations. Can anyone help? Is is even soluble?

Edit: Fixed a typo in the upper limit in the integral.

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    Unfortunately, the same exact substitution Davide points out shows that the RHS is the constant $\displaystyle \int_0^\infty f(s)ds.$ So nothing really changes here...2012-02-25

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Note that rhs of this equation is constant. Indeed, $ \int\limits_{-\infty}^{x} f(x-t)dt= \{\tau=x-t\}= \int\limits_{0}^{+\infty}f(\tau)d\tau= \int\limits_{0}^{+\infty}f(t)dt $ Therefore the lhs of this equation must be constant. But this is possible only if $a=0$. For the case when $a=0$, we have $1=\int\limits_{0}^{+\infty}f(t)dt$, otherwise there is no solution.

Finally if $a=0$ the solution of this equation is any integrable function $f$ such that $\int\limits_{0}^{+\infty}f(t)dt=1$. If $a\neq 0$, solution doesn't exist.

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    Ok, thank you! I think I made a deeper mistake in the problem that led me to this question. I'll dig tomorrow.2012-02-26