Question

(a) Let \(q(x) \leqslant 0\) in an interval \(I\). Prove that no solution of the equation $$\frac{d}{d x}\left(p \frac{d y}{d x}\right)+q y=0$$ can oscillate in \(I\). (b) Find the interval in which a nontrivial solution of the Legendre equation $$\left(1-x^{2}\right) y^{\prime \prime}-2 x y^{\prime}+2 y=0$$ has at most one zero.

Short answer

Question: Prove that no solution of the differential equation \(\frac{d}{dx}(p\frac{dy}{dx}) + qy = 0\) can oscillate in an interval \(I\) where \(q(x) \leq 0\), and find an interval in which a nontrivial solution of the Legendre equation \((1-x^2)y'' - 2xy' + 2y = 0\) has at most one zero. Solution: (a) No solution of the given differential equation can oscillate in an interval where \(q(x) \leq 0\). This is because, when we integrate the equation and analyze the constant of integration, we find that the constant \(C\) cannot be positive, which means no oscillation is possible in the interval. (b) In the interval \(x \in (-1,1)\), a nontrivial solution of the Legendre equation \((1-x^2)y'' - 2xy' + 2y = 0\) has at most one zero.

Step-by-step solution

Recall and rewrite the given equation

We start with the given equation \(\frac{d}{d x}\left(p \frac{d y}{d x}\right)+q y=0\).

Multiply both sides of the given equation by \(\frac{dy}{dx}\)

We will have \(\frac{d}{d x}\left(p \frac{d y}{d x}\right) \frac{d y}{d x} + q y \frac{d y}{d x} = 0 \frac{d y}{d x}\).

Integrate both sides of the equation with respect to \(x\)

We get \(\int \left(\frac{d}{d x}\left(p \frac{d y}{d x}\right) \frac{d y}{d x} + q y \frac{d y}{d x}\right) dx = \int 0 dx\).

Evaluate the integrals

The left side integral simplifies to \(\frac{1}{2}p\left(\frac{dy}{dx}\right)^2 + \frac{1}{2}qy^2 = C\), where \(C\) is the constant of integration, and the right side is a constant.

Show that the value of \(C\) cannot be positive

Since \(q(x) \leqslant 0\) and \(p(x) > 0\) in the interval \(I\), we have \(\frac{1}{2}p\left(\frac{dy}{dx}\right)^2 \geqslant 0\) and \(-\frac{1}{2}qy^2 \geqslant 0\). This implies that the sum \(\frac{1}{2}p\left(\frac{dy}{dx}\right)^2 - \frac{1}{2}qy^2 \leqslant 0\), which implies that \(C \leqslant 0\).

Conclude the proof

Since the constant \(C\) cannot be positive, no solution of the given differential equation can oscillate in the interval \(I\). (b) Find the interval with at most one zero:

Rewrite the given Legendre equation

We begin with the Legendre equation \(\left(1-x^{2}\right) y^{\prime \prime}-2 x y^{\prime}+2 y=0\).

Examine the Legendre equation behavior

The Legendre equation \(\left(1-x^{2}\right) y^{\prime \prime}-2 x y^{\prime}+\lambda y=0\) is well-known to describe functions with \(n\) zeros when \(\lambda=n(n+1)\). In our case, \(\lambda = 2\), so we should find a value of \(n\) that corresponds to at most one zero.

Find the value of n and the related interval

For \(\lambda=2\), we have \(n(n+1)=2\). The only integer value of \(n\) that satisfies this equation is \(n=1\). Therefore, in the interval \(x \in (-1,1)\), a nontrivial solution of the Legendre equation has at most one zero.

Key concepts

Oscillation Theorem

The Oscillation Theorem is a critical concept when dealing with ordinary differential equations (ODEs), especially concerning the behavior of solutions. It applies to equations of the form \[\frac{d}{d x}\left(p \frac{d y}{d x}\right)+q y=0,\]where the function \(q(x)\) is non-positive within a given interval \(I\). The theorem states that under these conditions, no solution of the differential equation can have an oscillatory behavior—meaning it cannot cross the x-axis multiple times in the interval \(I\). To understand why this is true, consider any solution \(y(x)\) of the equation. If \(y(x)\) were to oscillate, it would suggest a positive constant of integration after analyzing its integral form, which contradicts the requirement that \(C \leq 0\). Hence, the solutions are restricted to non-oscillating behavior as long as \(q(x) \leq 0\). This verification essentially prevents the function \(y(x)\) from having too many x-axis crossings within the interval, ensuring more stability in its shape.

Legendre Equation

The Legendre equation is a specific type of second-order differential equation and is a building block for solving various problems in mathematical physics. It is represented as:\[\left(1-x^{2}\right) y^{\prime \prime}-2 x y^{\prime}+2 y=0.\]This equation emerges in problems exhibiting spherical symmetry and is fundamental in solving Laplace's equation in spherical coordinates.Key characteristics of the Legendre equation include its solutions, known as Legendre polynomials or functions, which possess very intriguing behaviors. The solution structure is often analyzed in terms of their zeros, providing insights into their turns and oscillations over a defined interval. In practical terms, Legendre polynomials \(P_n(x)\) act as orthogonal polynomials in the interval \(-1 \le x \le 1\), a vital property utilized in various approximation algorithms and methods, like the Galerkin method in finite element analysis.Understanding the Legendre equation is essential when delving into special function theory, and it bridges many connections across different fields of physics and engineering, emphasizing the symmetric property of systems.

Boundary Conditions

When solving any differential equation, including the Legendre equation, understanding boundary conditions is crucial. These are constraints necessary to determine a unique solution among the infinity of possible solutions for differential equations.Boundary conditions often specify the value(s) that a solution must take on the boundary of the domain, or they express the rates of change of the solution. For instance:

• Dirichlet boundary conditions involve specifying the values at the boundary.
• Neumann boundary conditions define the derivative (or slope) at the boundary.

In the context of the Legendre equation, the natural boundaries of interest are the endpoints \(-1\) and \(1\) due to the nature of its argument \((1-x^2)\). Solving Legendre's equation with particular boundary conditions allows for the characterization of solution spaces, leading to unique, physically meaningful solutions in problems like potential flow and vibration analysis.Through combining these conditions with the differential equation, they serve as a guide to shaping the exact form of the solution, ensuring it fits the described physical or theoretical context.

Nontrivial Solution

In the context of differential equations, a nontrivial solution is a solution that is not identically zero. For example, in the Legendre equation:\[\left(1-x^{2}\right) y^{\prime \prime}-2 x y^{\prime}+2 y=0,\]seeking a nontrivial solution means finding a function \(y(x)\) that satisfies the equation without being the zero function everywhere. Nontrivial solutions are significant because they represent meaningful physical outcomes rather than the trivial state where nothing happens. When considering boundary conditions or an entire function space, the interest usually lies in these nontrivial solutions since trivial solutions often fail to encapsulate the problem's essence.In practice, identifying the interval over which a nontrivial solution has at most one zero involves solving conditions like \(n(n+1) = \lambda\). For our Legendre equation, where \(\lambda = 2\), this leads to \(n=1\), indicating an interval \(-1 < x < 1\) where nontrivial solutions exhibit limited zeros. This interval specifies where the solutions have a structured, predictable behavior related to the physical phenomena rightfully described by the differential equation.