$y\eta =\left (1-e^{(-p^2 v)}\right )y - ve^{x}\left(1-e^{-p^2y}\right)$
I would like to make $p$ the subject of the formula. Its seems a bit straightforward, but I have got stuck when I need to separate $e^{-p^2{v}}$ from $ve^{-p^2y + x}$. What manipulation can be done here?
I am afraid that it will not be possible in the most general case.
Let $z=e^{-p^2}$ and the expression is $$y\eta=(1-z^v)y-v e^x(1-z^y)$$ You could continue with $t=z^v$ and get $$y\eta=(1-t)y-v e^x(1-t^{\frac yv})$$
If $\frac yv$ is a small integer, you could solve as a polynomial.