Question

Consider the real eigenfunctions \(y_{n}(x)\) of a Sturm-Liouville equation $$ \left(p y^{\prime}\right)^{\prime}+q y+\lambda \rho y=0, \quad a \leq x \leq b $$ in which \(p(x), q(x)\) and \(\rho(x)\) are continuously differentiable real functions and \(p(x)\) does not change sign in \(a \leq x \leq b\). Take \(p(x)\) as positive throughout the interval, if necessary by changing the signs of all eigenvalues. For \(a \leq x_{1} \leq x_{2} \leq b\), establish the identity $$ \left(\lambda_{n}-\lambda_{m}\right) \int_{x_{1}}^{x_{2}} \rho y_{n} y_{m} d x=\left[y_{n} p y_{m}^{\prime}-y_{m} p y_{n}^{\prime}\right]_{x_{1}}^{x_{2}} $$ Deduce that if \(\lambda_{n}>\lambda_{m}\) then \(y_{n}(x)\) must change sign between two successive zeroes of \(y_{m}(x)\). (The reader may find it helpful to illustrate this result by sketching the first few eigenfunctions of the system \(y^{\prime \prime}+\lambda y=0\), with \(y(0)=y(\pi)=0\), and the Legendre polynomials \(P_{n}(z)\) given in subsection 16.6.1 for \(n=2,3,4,5\).)

Short answer

The identity is established by manipulating and integrating the differential equations. When \(\lambda_n > \lambda_m\), \(y_n(x)\) must change sign between successive zeros of \(y_m(x)\).

Step-by-step solution

Understand the given Sturm-Liouville equation

The given equation is \[ (p y')' + q y + \lambda \rho y = 0, \quad a \leq x \leq b \] where \(p(x)\), \(q(x)\), and \(\rho(x)\) are continuously differentiable, \(p(x)\) is positive and does not change sign in the interval.

Form the eigenfunctions and eigenvalues

Identify the real eigenfunctions \(y_n(x)\) and the corresponding eigenvalues \(\lambda_n\). For this, these functions must satisfy the given differential equation.

Set up the desired identity

We need to show the identity: \[ (\lambda_n - \lambda_m) \int_{x_1}^{x_2} \rho y_n y_m \, dx = \left[y_n \, p \, y_m' - y_m \, p y_n' \right]_{x_1}^{x_2} \]

Multiply the differential equations by appropriate functions

Multiply \( (p y_n')' + q y_n + \lambda_n \rho y_n = 0 \) by \(y_m\) and \( (p y_m')' + q y_m + \lambda_m \rho y_m = 0 \) by \(y_n\). This is to facilitate the elimination of terms later.

Subtract the resultant equations

Subtract the equation obtained from multiplying the second differential equation by \(y_n\) from that obtained from multiplying the first differential equation by \(y_m\).

Simplify the subtracted equation

After subtracting the two equations, we will integrate the result from \(x_1\) to \(x_2\). This should give the left-hand side of the desired identity.

Apply the integration by parts

The right-hand side of the final identity can be obtained using integration by parts and considering the boundary terms.

Use the conditions for eigenvalues \(\lambda_n > \lambda_m\)

To deduce the sign-changing behavior of \(y_n(x)\), note that if \(\lambda_n > \lambda_m\), the integral involving \(y_n\) and \(y_m\) must be non-zero, implying \(y_n(x)\) changes its sign between zeros of \(y_m(x)\).

Sketch the eigenfunctions as an illustration

For clarity, consider sketching the first few eigenfunctions for specific cases, e.g., for the system \(y'' + \lambda y = 0\) with boundary conditions \(y(0) = y(\pi) = 0\).

Key concepts

eigenfunctions

Eigenfunctions are special solutions to differential equations in the context of Sturm-Liouville theory. They arise when solving boundary value problems and satisfy the equation along with specific boundary conditions.
For example, the eigenfunctions, denoted as \(y_n(x)\), must satisfy the given differential equation:

\[ (p y')' + q y + \lambda \rho y = 0, \quad a \leq x \leq b \]
In this equation, \(p(x)\), \(q(x)\), and \(\rho(x)\) are functions of \(x\) that are continuously differentiable within the interval \(a \leq x \leq b\). Eigenfunctions typically come with corresponding eigenvalues that determine their specific form.
Understanding how to determine and work with eigenfunctions is critical when addressing problems involving differential equations, as it helps express solutions to various physical and mathematical problems.

eigenvalues

Eigenvalues, often denoted as \(\lambda_n\), represent constants in the Sturm-Liouville theory that accompany eigenfunctions. When you solve the Sturm-Liouville problem, you identify an infinite set of eigenfunctions, each corresponding to a unique eigenvalue.
More formally, these eigenvalues ensure that the differential equation and the boundary conditions are satisfied:

\[ (p y')' + q y + \lambda \rho y = 0, \quad a \leq x \leq b \]
These eigenvalues usually appear in ascending order and can impact the properties of the eigenfunctions. For instance, if you have two eigenvalues \(\lambda_n > \lambda_m\), their corresponding eigenfunctions \(y_n(x)\) and \(y_m(x)\) have specific relationships, such as the sign-changing behavior mentioned in the exercise solution.

boundary value problems

Boundary value problems (BVPs) are a common class of differential equations where you need to find a solution that satisfies certain conditions at the boundaries of the interval.
In the context of the Sturm-Liouville equation, these boundary conditions must be met at both ends of the interval \(a\) and \(b\).

For example, you may encounter boundary conditions like \(y(a) = 0\) and \(y(b) = 0\). These conditions restrict the set of possible solutions (eigenfunctions) and are crucial in determining the corresponding eigenvalues.
Properly handling boundary value problems is essential for accurate solutions in physical applications, such as vibrations, heat conduction, and quantum mechanics.

differential equations

Differential equations are mathematical equations involving derivatives, which describe how a function changes. They are fundamental to many areas of science and engineering.
The focus here is on the specific type known as the Sturm-Liouville equation:

\[ (p y')' + q y + \lambda \rho y = 0, \quad a \leq x \leq b \]
This second-order linear differential equation involves functions \(p(x)\), \(q(x)\), and \(\rho(x)\). Solving Sturm-Liouville problems involves finding the eigenfunctions and eigenvalues that satisfy both the differential equation and the boundary conditions.
Understanding differential equations and their properties help you model and solve real-world problems, such as population growth, heat distribution, and wave propagation.