1. The problem is to analyze and transform the partial differential equation (PDE) $$\frac{\partial^2 u}{\partial x^2} + e^{2x} \frac{\partial^2 u}{\partial y^2} + y \frac{\partial u}{\partial y} - x \frac{\partial u}{\partial x} = 0$$ and identify its type and canonical form. 2. Identify coefficients of the second derivatives: $$a_{11} = 1, \quad a_{12} = 0, \quad a_{22} = e^{2x}$$. 3. Calculate the discriminant $$D = a_{12}^2 - a_{11} a_{22} = 0^2 - 1 \cdot e^{2x} = -e^{2x} < 0$$, which means the PDE is elliptic. 4. The characteristic equation is $$a_{11} (dy)^2 - 2 a_{12} dx dy + a_{22} (dx)^2 = 0 \Rightarrow (dy)^2 + e^{2x} (dx)^2 = 0$$. 5. Since this has no real solutions, we proceed to find canonical variables by integrating the characteristic directions. 6. Given the substitution variables: $$z_1 = y + \frac{e^{2x}}{2}, \quad z_2 = y - x$$ 7. These transform the PDE into canonical form in variables $z_1, z_2$. 8. The PDE can be rewritten in terms of $V(z_1,z_2)$ with derivatives $V_{33}, V_{12}, V_{22}, V_{13}$ etc., as shown in the problem, leading to a simplified canonical PDE. 9. The key steps involve recognizing the elliptic nature, finding characteristic variables, and expressing the PDE in canonical form. Final answer: The PDE is elliptic with discriminant $$D = -e^{2x}$$ and canonical variables $$z_1 = y + \frac{e^{2x}}{2}, \quad z_2 = y - x$$ that simplify the PDE for further solution.