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Let u(x, t) be a smooth solution to the wave equation:

​$$\frac{\partial^2 u}{\partial t^2} - \frac{\partial^2 u}{\partial x^2} = 0, \quad \text{for } (x,t) \in \mathbb{R}^2$$​​

Which of the following is FALSE?

Consider the second-order PDE:

​$$a u_{xx} + b u_{xy} + a u_{yy} = 0 \quad \text{in } \mathbb{R}^2,$$

for $$a,b\in \mathbb{R}$$ . Which of the following is true? ​

​$$\text{Which of the following is a solution to }\\ u_x + x^2 u_y = 0 \text{ with } u(x, 0) = e^x?$$​​

​​$$\text{The general solution of the surfaces which are} \\ \text{perpendicular to the family of surfaces} \\z^2 = kxy, \; k \in \mathbb{R} \ \text{is}$$​​

Which of the following PDEs is NOT PARABOLIC for all x,y real numbers .

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Consider the wave equation for the wave equation :

Which one of the following is true ?

​The following partial differential equation is 

​ If u = u(x,t) is the solution of the initial value problem :

​ Let u = u(x,t) be the solution of the following initial value problem: