1. Let's start by stating the given equation: $$e^{x^2} y = e^y = x$$ It seems there's ambiguity in the expression as written because it contains two equal signs. Assuming the intended equation is either $$e^{x^2} y = e^y$$ or $$e^y = x$$, we'll analyze both cases. 2. Case 1: $$e^{x^2} y = e^y$$. We want to solve for $y$ in terms of $x$. 3. Rewrite the equation: $$e^{x^2} y = e^y$$. 4. Divide both sides by $e^{x^2}$: $$y = e^{y - x^2}$$. 5. Take the natural logarithm on both sides (valid if $y > 0$): $$\\ln y = y - x^2$$. 6. Rearrange: $$y - \\ln y = x^2$$. This is an implicit equation representing $y$ as a function of $x$. 7. This expression cannot be solved explicitly for $y$ with elementary functions but defines $y$ implicitly. 8. Case 2: From $$e^y = x$$, solve for $y$: 9. Take natural logarithm on both sides: $$y = \\ln x$$. Domain: $x > 0$. Final answers: - For the equation $$e^{x^2} y = e^y$$, $y$ satisfies the implicit relation $$y - \\ln y = x^2$$. - For the equation $$e^y = x$$, $y = \\ln x$ with $x > 0$.