Classify the following partial differential equation as elliptic, parabolic, or hyperbolic:

Classify the equation $x \frac{\partial^2 u}{\partial x^2} + y \frac{\partial^2 u}{\partial y^2} = 0$ in the different quadrants of the $xy$-plane.

Find the Fourier series for the square wave function $f(x)$ defined on $[-\pi, \pi]$:

Solve the 1D Heat Equation $\frac{\partial u}{\partial t} = \alpha^2 \frac{\partial^2 u}{\partial x^2}$ for a rod of length $L$, with boundary conditions $u(0,t) = 0$ and $u(L,t) = 0$, and an initial temperature distribution $u(x,0) = f(x)$.

A string of length $L=1$ and wave speed $c=2$ is fixed at both ends. It is plucked, giving it an initial displacement $u(x,0) = \sin(\pi x)$ and zero initial velocity $u_t(x,0) = 0$. Find the displacement $u(x,t)$.

Solve Laplace's equation $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$ for a rectangular plate defined by $0 \le x \le a$ and $0 \le y \le b$. Boundary conditions: $u(0,y) = 0$, $u(a,y) = 0$ (Left and right edges are 0) $u(x,0) = 0$ (Bottom edge is 0) $u(x,b) = f(x)$ (Top edge has a specified temperature distribution)