Question

The Laguerre \(^{11}\) differential equation is $$ x y^{\prime \prime}+(1-x) y^{\prime}+\lambda y=0 $$ Show that \(x=0\) is a regular singular point. Determine the indicial equation, its roots, the recurrence relation, and one solution \((x>0) .\) Show that if \(\lambda=m,\) a positive integer, this solution reduces to a polynomial. When properly normalized this polynomial is known as the Laguerre polynomial, \(L_{m}(x) .\)

Short answer

Question: Write down a solution for the given Laguerre differential equation $$xy'' + (1-x)y' + \lambda y = 0$$ and show how it reduces to a polynomial when λ=m. Answer: A solution for the given Laguerre differential equation when x > 0 can be represented as $$y(x) = a_0\left(1+\sum_{n=1}^{\infty} \frac{(1-\lambda)(1-\lambda-1)\cdots(1-\lambda-n+1)}{(n!)^2}x^n\right)$$. When λ=m, where m is a positive integer, the solution reduces to a polynomial. To obtain the normalized Laguerre polynomial, set a0 = 1, giving $$L_m(x) = 1+\sum_{n=1}^{m} \frac{(1-m)(1-m-1)\cdots(1-m-n+1)}{(n!)^2}x^n$$.

Step-by-step solution

Identify the singular points

Firstly, we will identify the singular points of the given Laguerre differential equation: $$ xy'' + (1-x)y' + \lambda y = 0 $$ Singular points are the points where the leading coefficient (in this case, `x`) vanishes. In our case, the singular point is at x=0.

Confirm the type of singular point

To check if x=0 is a regular singular point, we need to examine the following limit: $$ \lim_{x\to 0} (1-x)\frac{1}{x} = \lim_{x\to 0} \frac{1-x}{x} $$ The limit is finite since it equals 1, which confirms that x=0 is a regular singular point.

Determine the indicial equation and its roots

To determine the indicial equation, we will use the Frobenius method. Assuming a solution in the form of a power series: $$ y(x) = \sum_{n=0}^{\infty} a_nx^{r+n} $$ Plugging this into the differential equation and equating the coefficients of like powers of x, we get the indicial equation as: $$ r(r-1) + r + \lambda = 0 $$ Solving for r, we get the roots as: $$ r_1 = 0, \quad r_2 = 1 - \lambda $$

Find the recurrence relation

Now, we will derive the recurrence relation for the coefficients of the power series. Equating the coefficients of x^{r+n} for n≥1, we get: $$ a_{n+1} = \frac{\lambda - n}{(n+1)(n+r)} a_n $$

Construct one solution

Using the roots of the indicial equation and the recurrence relation, one solution for the given differential equation when x > 0 can be represented as: $$ y(x) = a_0\left(1+\sum_{n=1}^{\infty} \frac{(1-\lambda)(1-\lambda-1)\cdots(1-\lambda-n+1)}{(n!)^2}x^n\right) $$

Show the solution reduces to a polynomial when λ=m

If we let λ=m, where m is a positive integer, the numerator of the coefficient in the power series becomes: $$ (1-m)(1-m-1)\cdots(1-m-n+1) $$ When n = m, the numerator becomes zero, and all subsequent terms in the series vanish. Thus, the solution indeed reduces to a polynomial.

Normalize the polynomial as the Laguerre polynomial

To find the Laguerre polynomial Lm(x), normalize the polynomial by setting a_0 = 1: $$ L_m(x) = 1+\sum_{n=1}^{m} \frac{(1-m)(1-m-1)\cdots(1-m-n+1)}{(n!)^2}x^n $$ The polynomial Lm(x) is now properly normalized and is known as the Laguerre polynomial.

Key concepts

Singular Points

The concept of singular points is crucial in the study of differential equations, particularly when we're examining solutions near these points. Singular points are locations where the differential equation's coefficients become undefined or infinite, causing the behavior of the solutions to be less predictable. To identify a singular point, consider the leading coefficient of the highest derivative in the differential equation. Whenever this coefficient equals zero, you've located a singular point. In the context of the Laguerre equation, the singular point is identified at \( x=0 \), because the coefficient \( x \) in front of \( y''\) is zero there.

Regular Singular Point

A regular singular point is a subtler notion within singular points, requiring careful distinction. It must be checked whether the limits of certain expressions, derived from the differential equation when approaching the singular point, are finite. For instance, in the case of the Laguerre differential equation, the limit \[ \lim_{x\to 0} (1-x)\frac{1}{x} = \lim_{x\to 0} \frac{1-x}{x} \] is evaluated to check for regularity at the singular point \( x=0 \). Since this limit is finite, we confirm that \( x=0 \) is indeed a regular singular point. This classification has significant implications for finding solutions to the differential equation around this point using specialized methods such as the Frobenius method.

Indicial Equation

To understand an indicial equation, we first need to grasp the Frobenius method, which allows us to solve differential equations near a regular singular point using a power series approach. Applying this method, we search for solutions in the form of a power series expanded around the regular singular point. The coefficients of this power series are determined by what's called the indicial equation. For the Laguerre differential equation, after applying the Frobenius method and matching the coefficients of like powers of \( x \), the indicial equation \[ r(r-1) + r + \lambda = 0 \] emerges. Solving this equation gives us the roots \[ r_1 = 0, \quad r_2 = 1 - \lambda \], which are essential in defining the structure of the solution to the differential equation.

Frobenius Method

The Frobenius method is designed to find power series solutions in the neighborhood of regular singular points. It assumes the solution to be a power series whose initial point starts with the power \( r \) (where \( r \) is a root of the indicial equation). This method systematically calculates terms of the power series to build a solution. When utilizing the Frobenius method, we express the solution as \[ y(x) = \sum_{n=0}^{\infty} a_nx^{r+n} \] and substitute it into the differential equation. The ensuing calculations lead to the determination of the coefficients \( a_n \) through a recurrence relation.

Recurrence Relation

A recurrence relation is an equation that expresses each term of a sequence or array as a function of preceding terms. When solving differential equations with the Frobenius method, the recurrence relation provides a systematic way to compute the coefficients of the power series that constitutes a solution. In the context of the Laguerre equation, once the form of the solution is substituted, we obtain the relation \[ a_{n+1} = \frac{\lambda - n}{(n+1)(n+r)} a_n \], which enables us to calculate each coefficient based on the previously found coefficients, thus progressively building up the solution of the differential equation.

Laguerre Polynomial

Laguerre polynomials are a sequence of orthogonal polynomials that arise naturally when solving certain types of differential equations, including the Laguerre differential equation for specific values of \( \lambda \). If \( \lambda \) is taken to be a nonnegative integer, the power series solution truncates to a polynomial because the coefficient terms eventually vanish. For example, when \( \lambda = m \), where \( m \) is a positive integer, the series will terminate after \( m \) terms, giving rise to a Laguerre polynomial \[ L_m(x) \]. Normalizing the polynomial by setting the leading coefficient to 1 results in the standardized Laguerre polynomial \[ L_m(x) = 1+\sum_{n=1}^{m} \frac{(1-m)(1-m-1)\cdots(1-m-n+1)}{(n!)^2}x^n \], which plays a significant role in various areas of physics and mathematics.