Consider the equation
$$u_{xx}+2ayu_{xy}+e^{2x}u_{yy}-u=0$$
where $a$ is a real constant. Determine the value(s) of $a$ when this equation is elliptic everywhere in the $xy$-plane in which case find the canonical variables and reduce the equation to its canonical form.
I am trying to do this question for exam prep.
I got that the equation is elliptic when $(ay)^{2}-e^{2x} \lt 0$, that is that $a \lt \frac{e^{2x}}{y}$. However i'm not sure how i would determine the canonical variable and thus reduce the equation to canonical form. Any help is appreciated.
Your inequality is
$$ (ay - e^x)(ay + e^x) < 0 $$
which comes out to
$$ -e^x < ay < e^x $$
If $a\ne 0$, then the equation is ellliptic in the region bounded above and below by the curves $y = \pm \frac{e^x}{a}$. This is clearly not the entire $\mathbb R^2$ plane
Therefore $a=0$ is the only solution