The problem presented involves solving the wave equation:

$\frac{\partial^2 u}{\partial t^2} = 4 \frac{\partial^2 u}{\partial x^2}, \quad x \in (-\infty, \infty), \quad t > 0$ with the initial conditions: $u(x, 0) = 0, \quad u_t(x, 0) = x$

This is a classic second-order linear partial differential equation (PDE), often known as the 1-dimensional wave equation. The factor of 4 on the right-hand side implies a wave speed of $c = 2$, as the general form of the wave equation is:

$\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$

Solution Approach

The solution to the wave equation can be found using d'Alembert's formula for the general solution of the 1D wave equation. For $c = 2$, d'Alembert's solution is:

$u(x, t) = f(x - 2t) + g(x + 2t)$ where $f$ and $g$ are functions to be determined from the initial conditions.

Step 1: Apply the first initial condition $u(x, 0) = 0$

This gives: $u(x, 0) = f(x) + g(x) = 0 \quad \Rightarrow \quad f(x) = -g(x)$

Step 2: Apply the second initial condition $u_t(x, 0) = x$

The time derivative of the solution is: $u_t(x, t) = -2 f'(x - 2t) + 2 g'(x + 2t)$ At $t = 0$, this becomes: $u_t(x, 0) = -2 f'(x) + 2 g'(x) = x$ Using $f'(x) = -g'(x)$, we substitute: $-2 f'(x) + 2 (-f'(x)) = x \quad \Rightarrow \quad -4 f'(x) = x$ Thus, we solve for $f'(x)$: $f'(x) = -\frac{x}{4}$ Integrating this gives: $f(x) = -\frac{x^2}{8} + C$ Since $f(x) = -g(x)$, we also have $g(x) = \frac{x^2}{8} - C$.

Step 3: Form the general solution

The general solution is: $u(x, t) = f(x - 2t) + g(x + 2t)$ Substituting for $f$ and $g$: $u(x, t) = \left( -\frac{(x - 2t)^2}{8} + C \right) + \left( \frac{(x + 2t)^2}{8} - C \right)$ Simplifying: $u(x, t) = \frac{(x + 2t)^2 - (x - 2t)^2}{8}$ This expression simplifies further using the difference of squares: $u(x, t) = \frac{(x^2 + 4xt + 4t^2) - (x^2 - 4xt + 4t^2)}{8} = \frac{8xt}{8} = xt$

Final Solution

Thus, the solution to the wave equation is: $u(x, t) = xt$

Would you like any further clarifications or additional details on this solution?

Related Questions:

1. How is the wave speed $c$ derived from the wave equation?
2. What is d'Alembert's formula, and how is it used to solve wave equations?
3. How do the initial conditions influence the solution of a PDE?
4. Can this method of solving the wave equation be extended to higher dimensions?
5. How would the solution change if different initial conditions were applied?

Tip:

For wave equations, always check for symmetry in the initial conditions, as it can simplify finding $f(x)$ and $g(x)$ in d'Alembert's solution.