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I found the following differential equation:

$$\frac{\partial}{\partial t}f\left(x,t\right)=xf\left(x,t\right)+g\left(x\right)\int_{0}^{1}f\left(x',t\right)x'\mathrm{d}x'$$

to solve for $f(x,t)$. Here $g(x)$ is a given function of $x$, and we know the initial condition $f(x,0)$.

Before I try to solve this equation numerically, anyone knows if there is an analytical solution available?

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    $\begingroup$ It may be interesting to notice that $\frac{\partial}{\partial t}f\left(x,t\right)-xf\left(x,t\right)$ splits as a function of $x$ times a function of $t$. $\endgroup$ – Arnaud Mortier Jan 25 '18 at 18:25
  • $\begingroup$ @ArnaudMortier Yes, perhaps some form of variable separation can work. But I can't figure it out. $\endgroup$ – becko Jan 25 '18 at 19:22

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