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Partial Differential Equations (PDEs) are mathematical equations that involve rates of change with respect to continuous variables. Unlike ordinary differential equations, which involve derivatives with respect to a single variable, PDEs involve derivatives with respect to multiple independent variables. PDEs are essential in various fields such as physics, engineering, and finance because they model phenomena like heat conduction, sound waves, and the behavior of financial instruments.

PDEs can be linear or nonlinear, homogenous or inhomogenous. The equation in our textbook exercise, \( u_{xx} + u_{yy} = 0 \), is a homogeneous, linear PDE known as Laplace's equation, which is widely applied in electrostatics, fluid dynamics, and potential theory. Solving a PDE often involves specifying the boundary conditions, which our problem does as well. These boundary conditions are crucial as they describe how the solution behaves at the borders of the domain of interest.

Separation of Variables is a method used to solve PDEs where the solution is assumed to be the product of functions, each depending on a single coordinate direction. The idea is to reduce a PDE into multiple ordinary differential equations (ODEs) which can be solved individually.

In the textbook solution, the assumption is made that the solution \( u(x, y) \) can be represented as the product \( X(x)Y(y) \). This allows us to decouple the original PDE into simpler parts, making it more tractable. This form enables us to separate the two variables, x and y, and deal with them independently, vastly simplifying the problem.

The Fourier series is a way to represent a function as a sum of sinusoidal bases. It is particularly useful in solving PDEs because many physical systems are periodic in nature. When dealing with boundary value problems that involve functions defined on an interval, the Fourier series provides a powerful tool to decompose complex periodic functions into simpler sinusoidal components.

In our example, we use the Fourier sine series to express \( f(x) = x^2(\text{Ï€}- x) \), the function describing the boundary condition, in terms of its sinusoidal components. The coefficients \( B_n \) are calculated by integrating the product of \( f(x) \) and the sinusoidal terms, which describe the contribution of each component to the overall series. By doing so, we essentially project the function \( f(x) \) onto the basis functions, allowing for the reconstruction of the function within the boundary conditions provided.

Eigenvalue problems consist of finding nontrivial solutions of equations of the form \( L\textbf{u} = \text{λ}\textbf{u} \), where \( L \) is a linear operator, \( \textbf{u} \) is an unknown vector function, and \( \text{λ} \) represents eigenvalues. In the context of our PDE exercise, the eigenvalue problem stems from applying the boundary conditions to the separated equations of \( X(x) \) and \( Y(y) \). The boundary conditions transform the problem into one that seeks solutions that satisfy both the differential equation and the constraints at the boundaries.

Particularly, the conditions \( X(0) = 0 \) and \( X(a) = 0 \) allow us to find discrete eigenvalues \( \text{λ}_n \) and corresponding eigenfunctions \( X_n(x) = \text{sin}(\frac{n\text{π}}{a}x) \). These solutions are significant since they provide the modes by which the PDE solution behaves under the given spatial constraints. The eigenfunctions form a complete set, capable of representing the range of solutions within the domain of the problem.