Question

Transform each of the following equations into the equivalent selfadjoint form: (a) The Laguerre equation $$ x y^{\prime \prime}+(1-x) y^{\prime}+n y=0, \quad n=0,1,2, \ldots . $$ (b) The Hermite equation $$ y^{\prime \prime}-2 x y^{\prime}+2 n y=0, \quad n=0,1,2, \ldots . $$ (c) The Tchebycheff equation $$ \left(1-x^{2}\right) y^{\prime \prime}-x y^{\prime}+n^{2} y=0, \quad n=0,1,2, \ldots $$

Short answer

Question: Transform the following differential equations into their equivalent self-adjoint forms: (a) The Laguerre equation $$ x y^{\prime \prime}+(1-x) y^{\prime}+n y=0, \quad n=0,1,2, \ldots . $$ (b) The Hermite equation $$ y^{\prime \prime}-2 x y^{\prime}+2 n y=0, \quad n=0,1,2, \ldots . $$ (c) The Tchebycheff equation $$ \left(1-x^{2}\right) y^{\prime \prime}-x y^{\prime}+n^{2} y=0, \quad n=0,1,2, \ldots $$ Answer: (a) The equivalent self-adjoint form of the Laguerre equation is: $$ \dfrac{d}{dx}(x e^{-x} y^{\prime}) + n (x e^{-x})y = 0 $$ (b) The equivalent self-adjoint form of the Hermite equation is: $$ \dfrac{d}{dx}(e^{-x^2} y^{\prime}) + 2n (e^{-x^2})y = 0 $$ (c) The equivalent self-adjoint form of the Tchebycheff equation is: $$ \dfrac{d}{dx}\left(\dfrac{1}{\sqrt{1-x^2}} y^{\prime}\right) + n^2 \dfrac{1}{\sqrt{1-x^2}}y = 0 $$

Step-by-step solution

Identify weight function

To transform the given equation into its equivalent self-adjoint form, we need to identify a weight function, w(x), such that $$ \dfrac{d}{dx}(w(x)y'(x)) + n w(x)y(x) = 0 $$ Weight function can be determined by observing the coefficients of the given equation. In this case, the weight function is \(w(x) = x e^{-x}\).

Rewrite the equation using w(x)

Now that we have the weight function, we'll rewrite the given equation using w(x): $$ x y^{\prime \prime}+(1-x) y^{\prime}+n y = \dfrac{d}{dx}(x e^{-x} y^{\prime}) + n (x e^{-x})y $$ The equivalent self-adjoint form of the Laguerre equation is: $$ \dfrac{d}{dx}(x e^{-x} y^{\prime}) + n (x e^{-x})y = 0 $$ (b) The Hermite equation $$ y^{\prime \prime}-2 x y^{\prime}+2 n y=0, \quad n=0,1,2, \ldots . $$

Identify weight function

Again, we need to identify a weight function, w(x), such that $$ \dfrac{d}{dx}(w(x)y'(x)) + 2n w(x)y(x) = 0 $$ By observing coefficients of the given equation, the weight function in this case is \(w(x) = e^{-x^2}\).

Rewrite the equation using w(x)

Now that we have the weight function, rewrite the given equation using w(x): $$ y^{\prime \prime}-2 x y^{\prime}+2 n y = \dfrac{d}{dx}(e^{-x^2} y^{\prime}) + 2n (e^{-x^2})y $$ The equivalent self-adjoint form of the Hermite equation is: $$ \dfrac{d}{dx}(e^{-x^2} y^{\prime}) + 2n (e^{-x^2})y = 0 $$ (c) The Tchebycheff equation $$ \left(1-x^{2}\right) y^{\prime \prime}-x y^{\prime}+n^{2} y=0, \quad n=0,1,2, \ldots $$

Identify weight function

Again, we need to identify a weight function, w(x), such that $$ \dfrac{d}{dx}(w(x)y'(x)) + n^2 w(x)y(x) = 0 $$ Here, the weight function can be determined by observing the coefficients of the given equation. In this case, the weight function is \(w(x) = (1-x^2)^{-1/2}\).

Rewrite the equation using w(x)

Now that we have the weight function, rewrite the given equation using w(x): $$ (1-x^2) y^{\prime \prime}-x y^{\prime}+n^2 y = \dfrac{d}{dx}\left(\dfrac{1}{\sqrt{1-x^2}} y^{\prime}\right) + n^2 \dfrac{1}{\sqrt{1-x^2}}y $$ The equivalent self-adjoint form of the Tchebycheff equation is: $$ \dfrac{d}{dx}\left(\dfrac{1}{\sqrt{1-x^2}} y^{\prime}\right) + n^2 \dfrac{1}{\sqrt{1-x^2}}y = 0 $$

Key concepts

Laguerre Equation

The Laguerre equation is a second-order linear differential equation of major importance in applied mathematics and physics, particularly in the fields of quantum mechanics and orthogonal polynomials. It is expressed as

\[ x y'' + (1 - x) y' + n y = 0, \]
where \(y''\) denotes the second derivative of \(y\) with respect to \(x\), \(y'\) is the first derivative, and \(n\) is a non-negative integer representing the order of the associated Laguerre polynomial.

To transform this equation to self-adjoint form, a weight function \(w(x) = x e^{-x}\) is identified, such that

\[ \frac{d}{dx}(w(x)y'(x)) + n w(x)y(x) = 0. \]
This transformation not only simplifies the problem but also plays a crucial role in the orthogonality properties of the Laguerre polynomials, which are solutions to the equation.

Hermite Equation

The Hermite equation is another essential second-order linear differential equation, used extensively in probability theory and physics, especially in the description of the quantum harmonic oscillator. The equation is written as

\[ y'' - 2 x y' + 2 n y = 0, \]
with \(n\) as a non-negative integer. In this context, solutions to the Hermite equation are known as Hermite polynomials.

To find its self-adjoint form, the corresponding weight function is \(w(x) = e^{-x^2}\), which leads to

\[ \frac{d}{dx}(w(x)y'(x)) + 2n w(x)y(x) = 0. \]
The weight function \(e^{-x^2}\) is critical as it defines the Gaussian distribution's bell shape, reflecting the inherent properties of Hermite polynomials in the realm of probability and wave functions.

Tchebycheff Equation

The Tchebycheff (or Chebyshev) equation, named after the Russian mathematician Pafnuty Chebyshev, is fundamental to approximation theory and numerical analysis. It is given by

\[ (1 - x^2) y'' - x y' + n^2 y = 0, \]
where, as before, \(n\) is a non-negative integer. The Tchebycheff polynomials, which are solutions to this equation, are utilized for their minimax properties in polynomial approximation.

The self-adjoint form requires identifying the weight function \(w(x) = (1-x^2)^{-1/2}\), resulting in

\[ \frac{d}{dx}(w(x)y'(x)) + n^2 w(x)y(x) = 0. \]
The weight function is especially significant because it is related to the interval \([-1, 1]\) over which the Tchebycheff polynomials exhibit orthogonality, playing a vital role in their extensive use in numerical methods.

Weight Function

A weight function in the context of differential equations is a non-negative function that provides a measure for averaging or 'weighting' functions within an integral. Weight functions are integral to the study of orthogonal polynomials and the solution of self-adjoint differential equations.

For each of the aforementioned equations - Laguerre, Hermite, and Tchebycheff - the weight functions are distinct and are chosen so that when multiplied by the solution, they render the differential equation into a form that satisfies the property of self-adjointness. This property is essential because it ensures that the eigenvalues of the associated differential operator are real and the eigenfunctions (in this case, the polynomials) are orthogonal with respect to the weight function. In physical contexts, these functions often correspond to probability densities or distributions, determining how likely different outcomes are over a given range.