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For the equation below, I have found $y$ values of $y=\pm 1$ from multiplying $e$ across and dividing by $y$. I also know that $y=0$ but I can't seem to get the $y=0$ solution. How would I go about finding it? $$y\:e^{-y^2}\:=\:\frac{y}{e}$$

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    $\begingroup$ You should shift y/e to the other side and factorize y out. By dividing by y you lost a root, that is y = 0. $\endgroup$ – thedilated May 18 '17 at 7:16
  • $\begingroup$ @thedilated Thank you, I knew it was something simple like that $\endgroup$ – randb May 18 '17 at 7:19
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As per @thedilated's comment the correct way of solving this equation is to rewrite it as $$ye^{1-y^2} = y \iff y(e^{1-y^2}-1) = 0$$ so that either $y=0$ or $e^{1-y^2} = 1 \iff y^2 -1 =0$.

When you divide out by $y$, you are implicitly assuming that $y \neq 0$ since dividing by $0$ makes no sense, which is why you lost that solution. In general, when you divide both sides of an equation by variables, you need to make sure they are non-zero or deal with the possibility that they are $0$ seperately.

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