This is from an exam paper from last year that I’m going through for revision. It appears that many students who took this exam struggled with the question as the exam report says only 3 people got it correct. The question is:
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.
The part I’m struggling with is determining the region where the equation is elliptic. Obviously we need $a_{12}^2-a_{11}a_{22}<0$. So is the equation elliptic in the region $$-\frac{e^x}{y}<a<\frac{e^x}{y}$$ Also I’m struggling to work out what the characteristics of this equation are. Could someone give me some hints please? From there I should be able to figure this question out.