Question

Rewrite the equation in Sturm-Liouville form (with \(\lambda=0) .\) Assume that \(b, c, \alpha,\) and \(\nu\) are contants. $$ y^{\prime \prime}-2 x y^{\prime}+2 \alpha y=0 $$

Short answer

Question: Transform the given equation into Sturm-Liouville form with λ = 0: \(y''(x)-2xy'(x)+2\alpha y(x) = 0\). Answer: The Sturm-Liouville form with λ = 0 for the given equation is \(\frac{d}{dx}\left[x\frac{dy}{dx}\right] - y(x) = 0\) with \(p(x) = x\), \(q(x) = -1\), \(w(x) = 1\), and \(\lambda = 0\).

Step-by-step solution

Write down the given equation

The given equation is: $$ y''(x)-2xy'(x)+2\alpha y(x)=0 $$

Identify suitable function to multiply the equation

To rewrite the given equation in Sturm-Liouville form, we need to identify a suitable function to multiply the equation with. Let's try the function \(e^{-x^2}\).

Multiply the equation

Now, we multiply the given equation with the function \(e^{-x^2}\): $$ e^{-x^2}(y''(x)-2xy'(x)+2\alpha y(x)) = 0 $$ Or equivalently, $$ y''(x)e^{-x^2} - 2xy'(x)e^{-x^2} + 2\alpha y(x)e^{-x^2} = 0 $$

Integrate by parts

Let's integrate by parts because it may help us find the function in Sturm-Liouville form. Let the first term be \(u\) and the derivative of the second term be \(v^{\prime}\). Using the integration by parts formula \(u \int v^{\prime} dx = uv - \int v du dx\). Let \(u = y''(x)e^{-x^2}\) and \(v^{\prime} = 1\). Then \(v = x\), and, $$ u = e^{-x^2}y''(x) \Rightarrow du = (e^{-x^2}y''(x) + y'(x)(-2xe^{-x^2})) dx = (y''(x)e^{-x^2} - 2xy'(x)e^{-x^2}) dx $$ Now, integrate by parts: $$ \int du dx = y''(x)e^{-x^2} - 2xy'(x)e^{-x^2} = x\int v^{\prime} dx - \int v du dx = x - \int x(y''(x)e^{-x^2} - 2xy'(x)e^{-x^2}) dx $$ Rearranging: $$ \int x(y''(x)e^{-x^2} - 2xy'(x)e^{-x^2}) dx = x - y''(x)e^{-x^2} + 2xy'(x)e^{-x^2} $$

Transform into Sturm-Liouville form

Now we can rewrite the original equation using the result from Step 4: $$ x - y''(x)e^{-x^2} + 2xy'(x)e^{-x^2} + 2\alpha y(x)e^{-x^2} = 0 $$ So, the Sturm-Liouville form with λ = 0 is: $$ \frac{d}{dx}\left[x\frac{dy}{dx}\right] - y(x) = 0 $$ with \(p(x) = x, q(x) = -1, w(x) = 1\) and \(\lambda = 0\).

Key concepts

Differential Equations

Differential equations are mathematical expressions that describe relationships involving derivatives of an unknown function and the function itself. In its simplest form, a differential equation represents how a particular quantity changes with respect to another, such as time or space.

For example, in the given exercise, the equation is a second-order linear differential equation, which means it involves the second derivative of the function y with respect to the variable x, denoted as y''(x). The equation represents a relationship involving y, its first derivative y'(x), and its second derivative y''(x) along with the variable x and the constant \(\alpha\). Understanding differential equations is crucial for modeling countless physical phenomena, from the simple motion of objects to the complex dynamics of weather systems.

Integration by Parts

Integration by parts is a technique used to integrate products of functions and is derived from the product rule for differentiation. The formula is given by \( u \int v' dx = uv - \int v du dx \). By choosing \( u \) and \( v' \) wisely, you can simplify the integration process.

In the solution provided, integration by parts was employed to manipulate the original differential equation and extract a function in the Sturm-Liouville form. This is a strategic move often used in differential equations, especially when looking for special forms or simplifying expressions for further analysis.

Sturm-Liouville Theory

Sturm-Liouville theory is a theory of differential equations that enables the solving of complex boundary value problems. It often turns a difficult-to-solve differential equation into a more manageable form. A major aspect of Sturm-Liouville theory is the conversion of a regular differential equation into its unique Sturm-Liouville form, which can be characterized by a self-adjoint differential operator.

In the Sturm-Liouville form, such as the one derived from the given exercise, the equation involves a special kind of operator that, when applied to a function, yields an eigenvalue problem. The Sturm-Liouville form is an essential aspect of solving many physical problems, as it lays the groundwork for understanding eigenfunctions and eigenvalues, providing a systematic way to find solutions to a spread of complex problems.

Eigenvalue Problems

Eigenvalue problems are a type of problem in which an operator is applied to a function, and the result is proportional to the original function. This proportionality constant is known as the eigenvalue. Mathematically, it is often written in the form \( L[u] = \lambda w(x)u \), where L is a linear differential operator, \( w(x) \) is a weight function, u is the eigenfunction, and \( \lambda \) is the eigenvalue.

In the context of Sturm-Liouville problems, the eigenvalues can be considered as special values that arise in the solution of the equation. They often represent quantized states in physical systems, such as the energy levels in quantum mechanics or the natural frequencies of vibrating systems. Understanding eigenvalue problems is vital for solving many practical engineering and physics problems as they often yield solutions that form an orthogonal set, facilitating the expansion of functions into series for further analysis.