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The solution is often given by the method of characteristics which involves solving ordinary differential equations.

Analytic solutions are rare and often the problem is approached with numerical methods, especially finite difference or finite volume methods.

$u(x, t) = f(x - ct)$, where $f$ is an arbitrary function.

The solution often uses techniques from both wave equations and harmonic oscillator problems, and may require numerical approximation in complex scenarios.

Depending on boundary and initial conditions, methods like separation of variables or the Laplace transform can be used to approach the problem.

$u(x, t) = f(x - ct) + g(x + ct)$, where $f$ and $g$ are twice differentiable functions.

$u(\mathbf{x}) = \int_{\mathbb{R}^n} e^{i\mathbf{x} \cdot \mathbf{p}} \hat{f}(\mathbf{p}) e^{-k|\mathbf{p}|} d\mathbf{p}$, where $\hat{f}(\mathbf{p})$ is the Fourier transform of initial conditions.

$u(r, \theta) = A_0 + \sum_{n=1}^{\infty} r^n (A_n \cos(n\theta) + B_n \sin(n\theta))$, where $A_n$ and $B_n$ are constants.

$u(\mathbf{x}, t) = \frac{1}{(2\pi)^{n/2}} \int_{\mathbb{R}^n} e^{i\mathbf{x} \cdot \mathbf{p}} \hat{f}(\mathbf{p}) e^{-ict|\mathbf{p}|} d\mathbf{p}$, where $\hat{f}(\mathbf{p})$ is the Fourier transform of initial displacement.

$u(x, t) = \frac{1}{\sqrt{4\pi kt}}\int_{-\infty}^{\infty} e^{\frac{-(x-\xi)^2}{4kt}} \phi(\xi) d\xi$, where $\phi$ is the initial condition.

$u(x, t) = f(x - ut)$, where $f$ is the initial condition, and the solution is valid only until shock formation.

$u(r, t) = \frac{f(r - ct) + g(r + ct)}{r}$, where $f$ and $g$ are arbitrary functions, representing incoming and outgoing waves.

$u(x) = Ax^3 + Bx^2 + Cx + D$, where $A$, $B$, $C$, and $D$ are constants.

$u(z) = Ce^{kz} + De^{-kz}$, where $C$ and $D$ are constants.

Solution methods include separation of variables or the method of characteristics, applied according to specific boundary and initial conditions.

$u(x, y) = f(x + iy) + g(x - iy)$, where $f$ and $g$ are analytic functions.

A general solution might not exist in closed form and often the problem needs to be approached with numerical methods, such as finite elements or finite volumes.

Solution methods include separation of variables, spherical harmonics for radial symmetry, or using Green's function for specific boundary conditions.

$u(x, y, t) = f(x - t, y) + g(x + t, y)$, where $f$ and $g$ are arbitrary functions.

The general solution depends on the boundary conditions and is often found using separation of variables or integral transform methods.

$u(r) = \frac{C}{r}$, where $C$ is a constant.

The solution usually requires numerical methods like finite difference or finite element analysis, depending on the initial and boundary conditions.

The general solution depends on the boundary conditions and $D$, and is often tackled using numerical methods such as the finite element method.

$u(r) = \frac{C}{r^{n-1}}$, where $C$ is a constant and $n$ the dimension.

$u(x) = \frac{2k^2}{c} \mathrm{sech}^2\left( k(x - ct) - x_0 \right)$, where $k$, $c$, and $x_0$ are constants.

Solutions are obtained by transforming the variables to decouple the equation into simpler PDEs.

The solution involves the Green's function for the heat equation in three dimensions and will generally require numerical approximation methods.

The general solution will depend on boundary and initial conditions, typically requiring Green's functions or numerical methods.