Question

Rewrite the equation in Sturm-Liouville form (with \(\lambda=0) .\) Assume that \(b, c, \alpha,\) and \(\nu\) are contants. $$ x y^{\prime \prime}+(1-x) y^{\prime}+\alpha y=0 $$

Short answer

Based on the step-by-step solution, write a short answer to the question: To rewrite the given equation, \(x y^{\prime\prime}+(1-x) y^{\prime}+\alpha y=0\), in Sturm-Liouville form with \(\lambda=0\), we first identified the coefficients and their derivatives. Then, we divided the given equation by the function \(p(x) = x\), obtaining \(y^{\prime\prime}+\frac{1-x}{x} y^{\prime}+\frac{\alpha}{x} y = 0\). Finally, we integrated and found the Sturm-Liouville form to be: \(\frac{d}{dx} \left[x\frac{dy}{x dx}\right] - \alpha y = 0\).

Step-by-step solution

Identify the functions and their derivatives

In the given equation, we have the following functions and their derivatives: $$ p(x) = x, \quad p'(x) = 1, \quad q(x) = \alpha, \quad \lambda = 0 $$ The Sturm-Liouville form of the equation is given by $$ \frac{d}{dx} \left[p(x)\frac{dy}{dx}\right] + q(x)y = \lambda r(x)y. $$

Divide the equation by p(x)

To get a similar form as the Sturm-Liouville equation, we first need to divide the given equation by the function \(p(x)\): $$ \frac{1}{x}\left(x y^{\prime\prime}+(1-x) y^{\prime}+\alpha y\right) = y^{\prime\prime}+\frac{1-x}{x} y^{\prime}+\frac{\alpha}{x} y = 0. $$

Integrating to obtain the Sturm-Liouville form

Now we need to rewrite the equation by comparing the terms with the ones in the Sturm-Liouville equation. The term \(y^{\prime\prime}\) is already in the right place, so we need to focus on the first derivative term. Comparing the equation with the Sturm-Liouville form, we can observe that $ \frac{1-x}{x} y^{\prime} = \frac{d}{dx} \left[p(x)\frac{dy}{dx}\right] - q(x)y, $implies $$ \frac{d}{dx} \left[p(x)\frac{dy}{dx}\right] = \frac{1-x}{x} y^{\prime} + q(x)y. $$ Performing the differentiation and cancelling the terms, we get $$ \frac{d}{dx} \left[x\frac{dy}{x dx}\right] = \frac{1-x}{x} y^{\prime} + \alpha y. $$ Substituting back into the original equation, we get the Sturm-Liouville form of the given equation: $$ \frac{d}{dx} \left[x\frac{dy}{x dx}\right] - \alpha y = 0. $$

Key concepts

Differential Equations

Differential equations are mathematical equations that involve the rates at which quantities change. They are a core part of mathematics and are used to model various physical phenomena. In particular, a differential equation contains functions and their derivatives, which represent how a particular quantity varies with respect to another.
In the context of the given exercise, we observe the equation:

• \(x y^{\prime \prime}+ (1-x) y^{\prime}+ \alpha y = 0\).

Here, the unknown function is \(y(x)\), and its derivatives \(y'(x)\) and \(y''(x)\) represent the rate of change of \(y\) and the rate of change of the derivative of \(y\), respectively.
In many real-world problems, these equations help describe the dynamics of systems, such as motion, electricity, heat, and fluid flow.
Differential equations can be classified into ordinary differential equations (ODEs), which have a single independent variable, and partial differential equations (PDEs), which involve multiple independent variables. The given equation in the exercise is an ODE with respect to \(x\). Understanding how to work with these equations is crucial, as they form the foundation for many scientific and engineering disciplines.
By transforming differential equations into a form like the Sturm-Liouville form, we can leverage powerful solution techniques.

Boundary Value Problems

Boundary value problems (BVPs) are a special class of differential equations. They arise when we need to find a solution to a differential equation that satisfies certain conditions at the boundary of its domain. For physical problems, these boundaries can represent limitations in time or space.
For example, when analyzing how temperature changes along a rod, the temperatures at the two ends could be fixed. This would impose boundary conditions that must be satisfied by the solution of the differential equation describing the heat flow.
In BVPs, we often deal with linear, self-adjoint differential equations, which fit into the Sturm-Liouville theory. The given exercise focuses on rewriting the equation in Sturm-Liouville form, which facilitates the application of boundary condition analysis and allows us to find specific solutions that satisfy the boundaries.
These problems are a staple of engineering, physics, and mathematics because they model steady states and long-term behavior, rather than just instantaneous rates of change like initial value problems do. By solving BVPs, we learn about the constraints and potential behaviors of physical systems under specific conditions.

Sturm-Liouville Theory

Sturm-Liouville theory is a fundamental part of the study of differential equations, particularly dealing with linear second-order differential equations. This theory provides techniques for finding complete orthogonal sets of eigenfunctions which can be used to solve boundary value problems.
By expressing equations in Sturm-Liouville form, like the one in the exercise

• \(\frac{d}{dx} \left[x\frac{dy}{x dx}\right] - \alpha y = 0\),

we align the equation with a standard method that is known to yield solutions under specific boundary conditions.
The typical Sturm-Liouville problem includes a differential operator and boundary conditions imposed on the interval of interest. This method allows for elegant solutions using eigenvalues and eigenfunctions. The general form of a Sturm-Liouville equation is:
\[\frac{d}{dx}\left[p(x) \frac{dy}{dx}\right] + \left(q(x) - \lambda r(x)\right)y = 0,\]where \(p(x)\), \(q(x)\), and \(r(x)\) are given functions, and \(\lambda\) represents an eigenvalue.
This standardization not only makes the solutions tractable but also connects the theory to other important mathematical concepts like Fourier series, eigen-functions, and spectral theory. Understanding and applying Sturm-Liouville theory enables solving complex BVPs effectively, making it an indispensable tool in applied mathematics and physics.