List of top Partial Differential Equations Questions on Partial Differential Equations asked in GATE MA

Consider the second-order partial differential equation (PDE) \[ \frac{\partial^2 u}{\partial x^2} + 4 \frac{\partial^2 u}{\partial x \partial y} + (x^2 + 4y^2) \frac{\partial^2 u}{\partial y^2} = \sin(x + y) \] Consider the following statements: P: The PDE is parabolic on the ellipse \( \frac{x^2}{4} + y^2 = 1 \).
Q: The PDE is hyperbolic inside the ellipse \( \frac{x^2}{4} + y^2 = 1 \). Then:

If \( u(x, y) \) is the solution of the Cauchy problem \[ x \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} = 1, u(x, 0) = -x^2, x > 0, \] then \( u(2, 1) \) is equal to