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Any ideas how to solve this integral equation? $$f\left(x\right)\int_{-\infty}^{x}f\left(y\right)dy=\int_{-\infty}^{x}\int_{-\infty}^{x}\left(x-y\right)\left(x-z\right)f\left(y\right)f\left(z\right)dydz$$

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Let $F(x) = \int_{-\infty}^x(x-z)f(z)$. Then the equation becomes $$ F''(x)F'(x) = F^2(x) \, . $$ A family of solutions is given by $f(x) = ce^x$ for any $c \in \mathbb{R}$.

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