Question

Express the hypergeometric equation $$ \left(x^{2}-x\right) y^{\prime \prime}+[(1+\alpha+\beta) x-\gamma] y^{\prime}+\alpha \beta y=0 $$ in Sturm-Liouville form, determining the conditions imposed on \(x\) and on the parameters \(\alpha, \beta\) and \(\gamma\) by the boundary conditions and the allowed forms of weight function.

Short answer

The hypergeometric equation can be expressed in Sturm-Liouville form within the interval \((0,1)\) by defining the appropriate functions and ensuring parameter constraints such that the weight function is positive.

Step-by-step solution

- Identify the General Form

The Sturm-Liouville equation is given by \[ \frac{d}{dx}\bigg[p(x) \frac{dy}{dx}\bigg] + [\rho(x) - \beta \rho(x) r(x)]y = 0 \] where \(p(x)\) is a differentiable function, \(\rho(x)\) is a positive weight function, and \(r(x)\) is another function of x.

- Manipulate the Original Equation

Given the hypergeometric equation \[ (x^{2}-x) y^{''} + [(1+\beta)x-\rho] y^{'} + \beta \rho y = 0 \], divide through by \(x^2 - x\) to get it in the form: \[ y'' + \frac{(\beta + 1)x - \rho}{x^2 - x} y' + \beta \rho \frac{1}{x^2 - x} y = 0 \]

- Define New Functions

Identify the terms for the Sturm-Liouville form \[ \rho(x) = x - x^2 \], \[ p(x, y) = (1 + \beta)x - \rho \], and \[ q(x) = \rho \beta \] with their constraints and properties.

- Establish Conditions on Parameters

To ensure non-negativeness of the weight function and to avoid dividing by zero, place restrictions on \(x\), \(\beta\), and \(\rho\) such that \(0 < x < 1\).

- Determine the Weight Function

The weight function must be positive in the interval \((0,1)\), i.e., \(p(x) > 0\)

- Summarize Results

Under these conditions, the original hypergeometric equation can be written in the Sturm-Liouville form within the interval \((0,1)\) for the suitable chosen functions and parameters.

Key concepts

Understanding Hypergeometric Equation

The hypergeometric equation is a type of differential equation that can appear in many mathematical and physical problems. Its general form includes parameters α, β, and γ, which influence the equation's behavior:
\[ (x^2 - x) y'' + [(1 + \alpha + \beta)x - \gamma] y' + \alpha \beta y = 0 \]
This form is essential as it may be transformed to fit the Sturm-Liouville form, which simplifies analysis and solution methods. To achieve this, careful manipulation and understanding of each term's role are crucial.

Role of the Weight Function

In the context of differential equations, a weight function, denoted usually by \( \rho(x) \), is crucial. It must be positive over the given interval. For the hypergeometric equation when transformed to Sturm-Liouville form, the weight function is essential in ensuring the solution maintains physical and mathematical properties.
The weight function for our transformed form is:
\[ \rho(x) = x - x^2 \]
This needs to stay positive within the interval of interest, here (0,1). Therefore, we ensure the parameters and interval do not violate this requirement.

Delve into Differential Equations

Differential equations are mathematical equations that relate some function with its derivatives. They play a pivotal role in various fields like physics, engineering, and economics. In our problem, the Sturm-Liouville and hypergeometric equations are both second-order linear differential equations but with specific forms.
A standard second-order linear differential equation can be written as:
\[ a(x) y'' + b(x) y' + c(x) y = 0 \]
When converted to Sturm-Liouville form, it helps utilize orthogonal functions and weight functions for easier solution methodologies. This translation into the form involves leveraging the coefficients appropriately for both simplification and practical application of boundary conditions.

Boundary Conditions Explained

Boundary conditions are constraints required for the solutions of differential equations. These conditions make sure the solutions are applicable and meaningful for physical scenarios. For example, in a string vibration problem, boundary conditions could specify that the string's ends remain fixed.
In our problem's context, the boundary conditions ensure the Sturm-Liouville problem is well-posed by fixing the x-interval \[ 0 < x < 1 \] and ensuring the weight function remains positive. This is achieved by appropriately choosing the parameters \( \alpha, \beta, \gamma \), and \( x \) to maintain the equation's integrity and solvability within the specified domain.