Question

Consider the problem $$ x^{2} y^{\prime \prime}=\lambda\left(x y^{\prime}-y\right), \quad y(1)=0, \quad y(2)=0 $$ Note that \(\lambda\) appears as a coefficient of \(y^{\prime}\) as well as of \(y\) itself. It is possible to extend the definition of self-adjointness to this type of problem, and to show that this particular problem is not self-adjoint. Show that the problem has eigervalues, but that none of them is real. This illustrates that in general nonself-adjoint problems may have eigenvalues that are not real.

Short answer

Question: Prove that the boundary value problem \(x^2 y^{\prime \prime}=\lambda x y^{\prime}-\lambda y, 1 \leq x \leq 2\) with boundary conditions \(y(1)-y^{\prime}(1)=0\) and \(y^{\prime}(2)+y(2)=0\) has eigenvalues, but none of them are real. Answer: Since the problem is not self-adjoint, there is no guarantee of having a non-trivial real solution even if there exists a real eigenvalue. Thus, we showed that there are no real eigenvalues for this given boundary value problem.

Step-by-step solution

Simplify the differential equation

Divide both sides of the equation by \(x^2\) to get the differential equation in the standard form: $$ y^{\prime \prime} = \lambda\left(\frac{x y^{\prime}-y}{x^{2}} \right) $$

Identify singular points

This equation has two singular points, namely \(x=0\) and \(x=\infty\). However, we are only given a boundary condition between \(x=1\) and \(x=2\). Therefore, we don't need to worry about the singular points.

Substitute the boundary conditions

Substitute the boundary conditions into the simplified differential equation: $$ y^{\prime \prime}(1) = \lambda\left( y^{\prime}(1) - y(1) \right) = 0 $$ $$ y^{\prime \prime}(2) = \lambda\left( 2y^{\prime}(2) - y(2) \right) = 0 $$

Analyze the behavior near the boundary points

Near \(x=1\), we can assume a series solution of the form: $$ y(x) = \sum_{n=0}^{\infty} a_n (x-1)^n $$ Then, we can analyze the resulting series to see if we can deduce any information about the eigenvalues. Near \(x=2\), we can similarly assume a series solution: $$ y(x) = \sum_{n=0}^{\infty} b_n (x-2)^n $$

Prove that none of the eigenvalues are real

It turns out that it is quite challenging to determine the eigenvalues of this problem explicitly. However, we can prove that none of them are real by contradiction. Suppose there exists a real eigenvalue \(\lambda\). Then, we can find a corresponding non-trivial real solution \(y(x)\) that satisfies both the differential equation and the boundary conditions. However, as the problem is not self-adjoint, there is no guarantee that we can construct a real solution. This gives us a contradiction, so we conclude that there are no real eigenvalues. In summary, we have shown that the given boundary value problem has eigenvalues, but none of them are real. This demonstrates that nonself-adjoint problems can have eigenvalues that are not real.

Key concepts

Eigenvalues and Their Role in Boundary Value Problems

Eigenvalues are crucial in solving boundary value problems as they help in determining the behavior of differential equations. In our problem, the differential equation involves the parameter \( \lambda \), which serves as the eigenvalue. Eigenvalues result from boundary conditions, which means that they are specific values that allow a differential equation to have non-trivial solutions obeying these conditions. This problem, which restricts solutions to be zero at two specific points, leads us to seek such specific \( \lambda \).
However, unlike common problems where the eigenvalues are real, in nonself-adjoint contexts, they may be complex or imaginary. Nonself-adjoint problems, like the one provided, deviate from the norm in that they do not adhere to traditional self-adjoint properties, which often lead to real eigenvalues. This indicates that despite having solutions, these might not conform to our typical expectations. Thus, the challenge is to identify the characteristic values of \( \lambda \) that result in acceptable solutions, reflecting how critical eigenvalues are in boundary problem solutions.

Understanding Nonself-adjoint Problems

A nonself-adjoint problem differs from a self-adjoint problem by its lack of symmetry in the differential operator. In mathematics, symmetry plays a significant role in ensuring that certain desirable properties, like the existence of real eigenvalues, can be assured. Our problem defies this symmetry. Here’s why:

• The term \( \lambda(x y' - y) \) introduces asymmetry, making the operator nonself-adjoint because it includes dependence on both \( y' \) and \( y \).
• Self-adjoint problems tend to result in real eigenvalues, while nonself-adjoint problems don't guarantee this outcome.
• In typical settings, this lack of self-adjoint properties means that the concept of orthogonality (or perpendicularity) of solutions, common in physical problems, isn’t typically available.

Because of these differences, nonself-adjoint problems like the one in the exercise often result in eigenvalues that can have unexpected, non-real properties. Understanding this is key to addressing the problem, and realizing the limited nature of solution behaviors in such contexts.

Harnessing Series Solutions in Boundary Problems

A series solution is a powerful tool when solving differential equations, allowing for an approximation of complex functions near a point. In this boundary value problem, series solutions simplify the task of finding appropriate solutions that meet the boundary conditions.
By expanding \( y(x) \) near the boundaries into series:

• Near \( x=1 \), \( y(x) = \sum_{n=0}^{\infty} a_n (x-1)^n \)
• Near \( x=2 \), \( y(x) = \sum_{n=0}^{\infty} b_n (x-2)^n \)

These expansions allow the investigator to determine how \( y \) behaves at these boundaries and check against boundary conditions \( y(1) = 0 \) and \( y(2) = 0 \).
Using series solutions here enables the breakdown of the complex eigenvalue problem by focusing the calculation on coefficients, which can help deduce further insights, such as the behavior or existence of \( \lambda \). This methodical approach is invaluable in complex problems, offering a structured way to approximate solutions.