Consider the second order Partial Differential Equation $\textsf{(PDE)}$
$$4 x^{2} \frac{\partial^{2} u}{\partial x^{2}}+4(x+y) \frac{\partial^{2} u}{\partial x \partial y}+\left(x^{2}+y^{2}\right) \frac{\partial^{2} u}{\partial y^{2}}-u=0$$
Then which one of the following statements is correct?
- The $\textsf{PDE}$ is hyperbolic in the region $ \{(x, y) \in \mathbb{R}^{2}:-1 < x < 0, y < 0 \}$
- The $\textsf{PDE}$ is hyperbolic in the region $ \{(x, y) \in \mathbb{R}^{2}:-1 < x < \infty, y < 0 \}$
- The $\textsf{PDE}$ is elliptic in the region $ \{(x, y) \in \mathbb{R}^{2}:0 < x < 1, y > 0 \}$
- The $\textsf{PDE}$ is parabolic in the region $ \{(x, y) \in \mathbb{R}^{2}:1 < x < \infty, y < 0 \}$
