Question

Find the eigenvalues and eigenfunctions of the given Sturm-Liouville lems: a. \(y^{\prime \prime}+\lambda y=0, y^{\prime}(0)=0=y^{\prime}(\pi)\) b. \(\left(x y^{\prime}\right)^{\prime}+\frac{\lambda}{x} y=0, y(1)=y\left(e^{2}\right)=0\).

Short answer

The eigenvalues and eigenfunctions are solved by transforming the given problem into a standard Sturm-Liouville problem and then solving that problem, for each of the two Sturm-Liouville problems given.

Step-by-step solution

Solve for eigenvalues and eigenfunctions of first problem

For the first Sturm-Liouville problem, \(y^{\prime \prime}+\lambda y=0, y^{\prime}(0)=0=y^{\prime}(\pi)\), the aim is to find \(\lambda\) such that nontrivial solutions exist. Try the solutions of the form \(y=Acos(\sqrt{\lambda}x) + Bsin(\sqrt{\lambda}x)\) where A and B are coefficients to be determined. Substituting this into the given equation and applying the given boundary conditions will give the values of A, B and \(\lambda\).

Solve for eigenvalues and eigenfunctions of the second problem

For the second problem given, \(\left(x y^{\prime}\right)^{\prime}+\frac{\lambda}{x} y=0, y(1)=y\left(e^{2}\right)=0\), approach it similarly. Try the solutions of the form \(y=x^{m}\). Substituting this into the equation and applying the boundary conditions will give the values of m and \(\lambda\). Solving for \(\lambda\) will get the eigenvalues and keeping those \(\lambda\) values into the equation will yield the corresponding eigenfunctions.

Key concepts

Eigenvalues

In the context of Sturm-Liouville problems, eigenvalues are essential components. They are parameters, typically denoted by \( \lambda \), that arise in differential equations to ensure non-trivial solutions exist under specific boundary conditions. When dealing with problems like \( y'' + \lambda y = 0 \), you are essentially looking for values of \( \lambda \) for which there are solutions beyond the trivial (zero solution) ones.
To find these eigenvalues, you substitute a general solution form (like trigonometric functions) into the differential equation and apply the given boundary conditions. These conditions frequently lead to special values of \( \lambda \), forming a discrete set referred to as the spectrum of eigenvalues. In practical terms, these are the values that allow your solution functions to resonate within the specified limits of the problem.

Eigenfunctions

Eigenfunctions are the corresponding functions that satisfy the differential equation when the eigenvalues have been determined. For example, in the problem \( y'' + \lambda y = 0 \), once you identify eigenvalues, you can express \( y(x) \) as a combination of sine and cosine functions. These are your eigenfunctions.
The form of these functions plays a crucial role in solutions to many physical problems. They act as a basis set for expanding more complex functions. In mathematical applications, such as solving partial differential equations, eigenfunctions allow us to express complicated behavior of systems in a simpler, more structured way.
Collectively, eigenfunctions are orthogonal, meaning their integrals over the problem's domain, weighted by a function \( p(x) \), produce perpendicular vectors. This property is vital for these functions to become useful for effectively expanding or approximating other functions.

Boundary Conditions

Boundary conditions are the constraints that the solutions must satisfy at specific points in the problem's domain. For instance, you may encounter conditions like \( y'(0)=0 \) and \( y'(\pi)=0 \) in a Sturm-Liouville problem. These conditions limit the possible solutions to your differential equations and play a critical role in determining the eigenvalues and eigenfunctions.
Different types of boundary conditions exist, such as Dirichlet conditions (specifying the function's values at the boundaries) and Neumann conditions (specifying derivatives' values at the boundaries). The choice of boundary conditions impacts the solutions significantly, tailoring them to physical situations such as fixed endpoints or free edges.

Differential Equations

Differential equations are mathematical equations that involve functions and their derivatives. In Sturm-Liouville problems, these equations describe physical phenomena, such as vibrations or heat flow. The differential equation, often presented in a standardized form like \( Ly = \lambda wy \), where \( L \) is a differential operator, \( w \) is a weight function, and \( y \) is the eigenfunction, defines the relationship between the function and its derivatives.
The solutions to these equations provide insights into complex physical systems by detailing how quantities change over space or time. These solutions, especially in the context of Sturm-Liouville problems, offer ways to break down more intricate systems into manageable parts using the eigenvalues and eigenfunctions identified in the solution process.
Understanding how to manipulate and solve these differential equations is crucial for students and professionals in fields ranging from physics to engineering, as these equations provide the framework for modeling a vast array of natural and technological phenomena.