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Boundary-value problems are mathematical problems in which we seek to find a function that satisfies a differential equation and meets specific conditions at the boundaries of the domain. In the given exercise, we are dealing with a problem involving the wave equation, where the function \( u(x, t) \) must satisfy conditions at both ends, \( x = 0 \) and \( x = 1 \). These boundary conditions, \( u(0, t) = 0 \) and \( u(1, t) = 0 \), mean that the displacement is zero at the ends of the domain for all time \( t > 0 \). This constraints the form of our solution and leads us to use methods like Fourier sine series, which inherently fit these zero-boundary conditions.

The wave equation is a second-order partial differential equation that describes the propagation of various types of waves, such as sound, light, and water waves. It is generally expressed as \( \frac{\partial^{2} u}{\partial x^{2}} = \frac{\partial^{2} u}{\partial t^{2}} \). In this problem, the wave equation models the behavior of a string fixed at both ends. Here, \( u(x, t) \) represents the displacement of the wave at position \( x \) and time \( t \). Understanding the wave equation is crucial because it helps us analyze how waves evolve over time, and predict future wave behavior based on initial and boundary conditions.

Separation of variables is a common technique for solving partial differential equations like the wave equation. The method assumes that a solution can be expressed as a product of functions, each a function of a single variable. For this exercise, we let \( u(x, t) = X(x)T(t) \), allowing us to separate the equation into two ordinary differential equations. This transformation simplifies solving complex problems, because it breaks the initial problem into more manageable pieces: one depending only on \( x \) and the other only on \( t \). This technique elegantly aligns with the Fourier series approach by providing functions that naturally satisfy the boundary conditions.

Initial conditions specify the state of the wave at \( t = 0 \), providing crucial information for its future evolution. In this problem, \( u(x, 0) = \sin(\pi x) \) sets the initial wave profile, indicating how the string is first displaced. Additionally, alongside the boundary conditions, we have the first time derivative \( \frac{\partial u}{\partial t}\big|_{t=0} = -\sin(\pi x) \), describing the initial velocity of the wave. These conditions ensure that our final solution not only obeys the boundary constraints but also reflects the wave's starting conditions accurately. The initial conditions are essential to determine the coefficients in the solution series, which specify how each mode or harmonic contributes to the complete wave solution.