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I have come across an equation of the following form $$\frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}(y-x)^{2}}[C_{1}x+C_{2}y+C_{3}x^{2}y + C_{3}y^{2}x] = C_{4}xy$$ where $C_{j}$ are real-valued constants. I'd like to be able to solve for $x$ in closed form, in terms of $y$ and the $C_{j}$. Alternately, I would also like to show the following result (which I am not sure holds): given $C_{1}<C_{2}$, show that for any $x,y$ satisfying this equation, $x>y$. Any help is greatly appreciated.

Edit: I mean the constants to be arbitrary except for the one restriction mentioned, so I am looking for a general solution in that sense.

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up vote 1 down vote accepted

If $C_1=C_3=0$ and $C_2=1$ (so $C_1\lt C_2$), the equation simplifies to $${\rm stuff\ }=C_4x$$ so $x$ can be pretty much anything just by choosing $C_4$ appropriately. In particular, $x\le y$ is not ruled out.

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stuff is what? please explain. – Victor Mar 30 '12 at 3:19
I mean the constants to be arbitrary with no restrictions except what I mentioned. So although this is true for a particular case I hope to solve in general. – Red Rover Mar 30 '12 at 3:21
If $x\gt y$ is false in a particular case, then you're not going to be able to show that it's true in general, are you? – Gerry Myerson Mar 30 '12 at 4:36
Anyway, you're not going to be able to solve that equation for $x$ in closed form. It's not even possible to solve $xe^x=2$ in closed form in terms of exponentials, logs, trig functions, powers, square roots, etc. – Gerry Myerson Mar 30 '12 at 4:39

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