Question

In Example 13.2 .4 we found that the eigenvalue problem $$ x^{2} y^{\prime \prime}+x y^{\prime}+\lambda y=0, \quad y(1)=0, \quad y(2)=0 $$ is equivalent to the Sturm-Liouville problem $$ \left(x y^{\prime}\right)^{\prime}+\frac{\lambda}{x} y=0, \quad y(1)=0, \quad y(2)=0 $$ Multiply the differential equation in (B) by \(y\) and integrate to show that $$ \lambda \int_{1}^{2} \frac{y^{2}(x)}{x} d x=\int_{1}^{2} x\left(y^{\prime}(x)\right)^{2} d x $$ Conclude from this that the eigenvalues of (A) are all positive.

Short answer

Question: Show that the eigenvalues of the given Sturm-Liouville problem are all positive. Given Sturm-Liouville problem: $$ (xy'(x))' + \frac{\lambda}{x} y(x) = 0 $$ Solution: We have shown that after multiplying the equation by y(x), integrating over the interval [1, 2], separating the integrals, using integration by parts, and simplifying the equation, we arrived at the final equation: $$ \int_{1}^{2} x(y'(x))^{2} \, dx = \lambda \int_{1}^{2} \frac{y^{2}(x)}{x} \, dx $$ Since the left side of the equation is always positive, it's clear that the eigenvalues on the right side (λ) must also be positive. Thus, we conclude that the eigenvalues of the given problem are all positive.

Step-by-step solution

Multiply the equation by y(x)

First, multiply the given Sturm-Liouville problem by the function y(x): $$ y(x) \left[ (xy'(x))' + \frac{\lambda}{x} y(x) \right] = 0 $$

Expand and simplify

Expand the equation and simplify: $$ (xy'(x))' y(x) + \frac{\lambda y^{2}(x)}{x} = 0 $$

Integrate both sides

Integrate both sides of the equation over the interval [1, 2]: $$ \int_{1}^{2} [(xy'(x))' y(x) + \frac{\lambda y^{2}(x)}{x}]\, dx = 0 $$

Separate the integrals

Separate the integrals into two parts: $$ \int_{1}^{2} (xy'(x))' y(x) \, dx + \lambda\int_{1}^{2} \frac{y^{2}(x)}{x} \, dx = 0 $$

Move one integral to the other side

Move the second integral to the right side of the equation: $$ \int_{1}^{2} (xy'(x))' y(x) \, dx = -\lambda \int_{1}^{2} \frac{y^{2}(x)}{x} \, dx $$

Integration by Parts for the Left Side Integral

Use integration by parts to integrate the left side of the equation. Let \(u=y(x)\) and \(dv=(xy'(x))' dx\). Thus, \(du=y'(x)dx\), \(v=x\). Now we can integrate by parts: $$ \int u\, dv = uv - \int v\, du = xy(x)y'(x) \Big|_{1}^{2} - \int_{1}^{2} xy'(x)y'(x) \, dx $$ The term \(xy(x)y'(x) \Big|_{1}^{2}\) equals zero since \(y(1) = 0\) and \(y(2) = 0\). Now, rewrite the equation: $$ \int_{1}^{2} (xy'(x))' y(x) \, dx = -\int_{1}^{2} x(y'(x))^{2} \, dx $$

Divide both sides by the Negative Sign

Divide both sides of the equation by -1 to get the final equation: $$ \int_{1}^{2} x(y'(x))^{2} \, dx = \lambda \int_{1}^{2} \frac{y^{2}(x)}{x} \, dx $$ Since the left side of the equation is always positive, it's clear that the eigenvalues on the right side (λ) must also be positive. Thus, we conclude that the eigenvalues of the given problem are all positive.

Key concepts

Eigenvalue Problem

When dealing with an eigenvalue problem, we're exploring a special type of equation. An eigenvalue problem aims to find values (called eigenvalues) for which there exists a non-trivial solution (called an eigenfunction) to a differential equation. These problems commonly occur in physics and engineering, such as vibrations or quantum mechanics.

In the given exercise, the original eigenvalue problem is expressed as:

• \(x^{2} y^{\prime \prime}+x y^{\prime}+\lambda y=0\)

The goal is to find the values of \(\lambda\) such that a non-zero solution \(y\) satisfies the given boundary conditions \(y(1) = 0\) and \(y(2) = 0\).

The eigenvalue problem is tackled by proving that the eigenvalues \(\lambda\) are positive. We do this by manipulating and integrating the problem using the skills in integration and differential equations to arrive at a condition that ensures positivity.

Integration by Parts

Integration by parts is a powerful technique used to simplify complex integrals. Derived from the product rule for differentiation, this method helps in transforming the product of functions into a more manageable form. For an integral \(\int u \, dv\), integration by parts is represented as:

\[\int u \, dv = uv - \int v \, du\]

In our exercise, we apply integration by parts to the integral \(\int_{1}^{2} (xy'(x))' y(x) \, dx\).

We choose:

• \(u = y(x)\)\
• \(dv = (xy'(x))' \, dx\)

Then we derive:

• \(du = y'(x) \, dx\)
• \(v = x\)

This approach helps simplify the integration to:

• \(xy(x)y'(x) \Big|_{1}^{2} - \int_{1}^{2} xy'(x)y'(x) \, dx\)

The boundary terms, \(xy(x)y'(x) \Big|_{1}^{2}\), vanish due to the boundary conditions, greatly simplifying our work.

By integrating this way, the originally daunting integral is broken down, revealing the relationships between parts of the problem that are critical to solving the eigenvalue problem.

Differential Equations

Differential equations are mathematical equations that involve an unknown function and its derivatives. They are essential in describing various natural and physical phenomena, such as heat, sound, dynamics, and more.

In our discussion, the differential equation is:

• \(x^{2} y^{\prime \prime}+x y^{\prime}+\lambda y=0\)

Which, after manipulation, can transform into another equivalent form:

• \(\left(x y^{\prime}\right)^{\prime}+\frac{\lambda}{x} y=0\)

This transformation allows us to employ the tools of Sturm-Liouville theory.

This theory helps in transforming the problem into a form that permits the application of eigenvalue techniques, such as proving the positivity of \(\lambda\) through integration. Solving differential equations like the one here involves mastering techniques like separation of variables, integrating factors, and boundary condition application. These tools are indispensable for comprehending complex systems described by differential equations.