Show that there exists a neighbourhood $U$ of $0$ in $\mathbb{R}$ and an unique $C^{\infty}$ function $g:U \to \mathbb{R}$ such that $g(0)=e$; and $\forall x \in U$,$x^2e^{g(x)}+g(x)^2e^x=1$.
Defining $f(x,y)=x^2e^{y}+y^2e^x$, I have that $\frac{\partial f}{\partial y}(0,e)=(x^2e^y+2ye^x)_{|(0,e)}=2e \neq 0$, so by Implicit Function Theorem, there exists $U$ neighborhood of $0$ in $\mathbb{R}$ such that as $f \in C^{\infty}$, there exists a $C^{\infty}$ function $g: U \to \mathbb{R}$ such that $y=g(x)$ for $x \in U$. But now how should I proceed to prove its uniqueness?