Problem 51
Question
Find the first three positive values of \(\lambda\) for which the proble $$ \begin{aligned} &\left(1-x^{2}\right) y^{\prime \prime}-2 x y^{\prime}+\lambda y=0, \\ &y(0)=0, y(x), y^{\prime}(x) \text { bounded on }[-1,1] \end{aligned} $$ has nontrivial solutions.
Step-by-Step Solution
Verified Answer
The first three positive eigenvalues are \(\lambda_1 = 2\), \(\lambda_2 = 12\), and \(\lambda_3 = 30\).
1Step 1: Identify the Type of Equation
The given differential equation \((1-x^2)y'' - 2xy' + \lambda y = 0\) is recognized as a Legendre differential equation. It often appears in problems involving eigenvalues.
2Step 2: Set Boundary Conditions
The boundary conditions are given as \(y(0) = 0\) and the requirements that \(y(x)\) and \(y'(x)\) must be bounded on \([-1, 1]\). These conditions imply that solutions are polynomials, specifically Legendre polynomials.
3Step 3: Use Properties of Legendre Polynomials
Legendre polynomials \(P_n(x)\) satisfy the equation for specific values of \(\lambda\) given by \(\lambda = n(n+1)\). The solutions are polynomials of degree \(n\).
4Step 4: Find the First Nontrivial Solutions
For nontrivial solutions, we need \(P_n(x)\) where \(n\) ensures non-zero solutions under given conditions. \(y(0)=0\) implies that \(P_n(0)=0\), which holds when \(n\) is odd.
5Step 5: Calculate First Three Positive Eigenvalues
Using \(\lambda = n(n+1)\) and noting the odd conditions for \(n\), substitute \(n=1\), \(3\), and \(5\). The corresponding eigenvalues are \(\lambda_1 = 2\), \(\lambda_2 = 12\), and \(\lambda_3 = 30\).
Key Concepts
EigenvaluesBoundary ConditionsLegendre Polynomials
Eigenvalues
In mathematical terms, eigenvalues are special numbers associated with a system of equations. In the context of differential equations, they help determine the solutions that respect given constraints. Here, the differential equation involves eigenvalues as one of its core components. The Legendre differential equation, a well-known form of second-order linear differential equation, incorporates a parameter \( \lambda \), which acts as an eigenvalue. These values are critical for finding nontrivial solutions to the equation.
To determine these eigenvalues, we utilize properties of Legendre polynomials, noting that for this equation, they adhere to the expression \( \lambda = n(n+1) \). This formula helps pinpoint the specific eigenvalues that satisfy the equation with polynomial solutions.
To determine these eigenvalues, we utilize properties of Legendre polynomials, noting that for this equation, they adhere to the expression \( \lambda = n(n+1) \). This formula helps pinpoint the specific eigenvalues that satisfy the equation with polynomial solutions.
Boundary Conditions
Boundary conditions are essential to solving differential equations as they define the limitations within which solutions must exist. In this particular problem, the boundary conditions are \( y(0) = 0 \) and the requirement that \( y(x) \) and its derivative \( y'(x) \) remain bounded within the interval \([-1, 1]\). These conditions ensure that the solutions conform to a certain behavior within the given range.
Due to these boundary conditions, the solutions are restricted to be polynomials that are not only bounded but also have specific properties at the endpoints. Specifically, this often means working with Legendre polynomials, which naturally satisfy the given conditions and boundary restraints. Such constraints are crucial in defining the types of solutions that can be considered valid for the differential equation.
Due to these boundary conditions, the solutions are restricted to be polynomials that are not only bounded but also have specific properties at the endpoints. Specifically, this often means working with Legendre polynomials, which naturally satisfy the given conditions and boundary restraints. Such constraints are crucial in defining the types of solutions that can be considered valid for the differential equation.
Legendre Polynomials
Legendre polynomials, famously associated with the solutions to the Legendre differential equations, are a set of orthogonal polynomials indexed by degree \( n \). They are particularly significant in solving equations involving spherical systems and have applications in physics and engineering.
These polynomials are derived to satisfy certain differential equations like the one given here, and they occur naturally when solving problems with specific symmetry and boundary conditions. In this exercise, Legendre polynomials arise because the boundary condition \( y(0) = 0 \) and the requirement for bounded solutions imply natural connections to these polynomials.
For odd \( n \), Legendre polynomials ensure that the polynomial value at 0 is zero, making them suitable under the equation’s conditions. Thus, by finding the polynomials \( P_n(x) \) for odd values of \( n \), we identify acceptable solutions, leading us directly to the eigenvalues \( \lambda \) through \( \lambda = n(n+1) \). Each odd \( n \) gives one positive and valid eigenvalue, as calculated in the original problem's solution.
These polynomials are derived to satisfy certain differential equations like the one given here, and they occur naturally when solving problems with specific symmetry and boundary conditions. In this exercise, Legendre polynomials arise because the boundary condition \( y(0) = 0 \) and the requirement for bounded solutions imply natural connections to these polynomials.
For odd \( n \), Legendre polynomials ensure that the polynomial value at 0 is zero, making them suitable under the equation’s conditions. Thus, by finding the polynomials \( P_n(x) \) for odd values of \( n \), we identify acceptable solutions, leading us directly to the eigenvalues \( \lambda \) through \( \lambda = n(n+1) \). Each odd \( n \) gives one positive and valid eigenvalue, as calculated in the original problem's solution.
Other exercises in this chapter
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