Question

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

Short answer

Question: Rewrite the given differential equation in Sturm-Liouville form with λ = 0: $$ x^2y'' + xy' + (x^2 - \nu^2)y = 0 $$ Answer: The Sturm-Liouville form of the given equation is: $$ \frac{d}{dx}\left[x^2 \frac{dy}{dx}\right] + (x^2 - \nu^2)y = 0 $$ Where p(x) = x², q(x) = x² - ν², and r(x) = 1.

Step-by-step solution

Matching terms with Sturm-Liouville form

Let's notice that which terms in the Sturm-Liouville form match with the given equation: For the first term in the Sturm-Liouville form: $$ \frac{d}{dx}\left[p(x) \frac{dy}{dx}\right] \Rightarrow x^{2} y^{\prime \prime}+x y^{\prime} $$ For the second term: $$ q(x)y \Rightarrow (x^2 - \nu^2)y $$ Now let's find p(x), q(x), and r(x) functions.

Identify p(x)

To identify p(x), we will look at the first term of the Sturm-Liouville form, which is: $$ \frac{d}{dx}\left[p(x) \frac{dy}{dx}\right] $$ The equation is already given in the form where we can identify p(x): $$ x^{2} y^{\prime \prime} + x y^{\prime} $$ We can rewrite the above equation as: $$ \frac{d}{dx}\left[x^{2} \frac{dy}{dx}\right] $$ Therefore, p(x) is: $$ p(x) = x^2 $$

Identify q(x)

To identify q(x), we will look at the second term of the Sturm-Liouville form, which is: $$ q(x) y \Rightarrow (x^2 - \nu^2) y $$ Comparing this with the given equation term (x² - ν²)y, we can directly identify q(x) as: $$ q(x) = x^2 - \nu^2 $$

Identify r(x) and write the final equation

Since λ = 0, r(x) doesn't have an effect on the equation, but for the sake of completeness, let's identify r(x). In the Sturm-Liouville form, r(x)y term should include lambda: $$ \lambda r(x)y $$ Since λ=0, r(x) can be any function, for example: $$ r(x) = 1 $$ Now we can rewrite the original equation in Sturm-Liouville form: $$ \frac{d}{dx}\left[x^2 \frac{dy}{dx}\right] + (x^2 - \nu^2)y = 0 $$ Where p(x) = x², q(x) = x² - ν², and r(x) = 1.

Key concepts

Differential equations

Differential equations are equations that involve a function and its derivatives. They play a crucial role in describing physical phenomena such as motion, heat, light, and more. In general, solving a differential equation means finding the function or set of functions that satisfy the equation.In our particular example, we have a second-order differential equation given by:\[ x^2 y'' + x y' + (x^2 - u^2) y = 0 \]This type of equation is common in physics and engineering, and requires specific methods for solution depending on its form and constraints. Differential equations can be quite complex, so it's important to identify their order (the highest derivative that appears) and type. This helps determine the best approach to find their solutions.

Sturm-Liouville form

The Sturm-Liouville form is a specific way of writing linear second-order differential equations that helps in solving them using standard methods. It is particularly useful in problems involving eigenvalues and eigenfunctions, which are important concepts in many areas of physics and engineering.An equation is in Sturm-Liouville form if it can be expressed as:\[ \frac{d}{dx}\left[ p(x) \frac{dy}{dx} \right] + q(x)y = \lambda r(x)y \]Here, \( p(x) \), \( q(x) \), and \( r(x) \) are known functions, and \( \lambda \) is a parameter often related to eigenvalues. By rewriting our given equation in this form with \( \lambda = 0 \), we simplify the process of finding its solutions while preserving essential properties of the function \( y \).

Eigenvalue problems

Eigenvalue problems in differential equations involve finding characteristic values, called eigenvalues, for which the differential equation has non-trivial solutions, known as eigenfunctions.These problems often arise in the context of the Sturm-Liouville form of differential equations, where \( \lambda \) represents the eigenvalue. Solutions to such problems are significant in various scientific fields as they can represent modes of vibration, frequencies, or energy levels in quantum mechanics.Though the step-by-step solution in this exercise assumes \( \lambda = 0 \), making it a special case, it highlights the general method of how these problems are studied: by first identifying the standard form, and then resolving or simplifying the equation to discover relevant eigenvalues and eigenfunctions.

p(x), q(x), r(x) identification

Identifying the functions \( p(x) \), \( q(x) \), and \( r(x) \) is crucial to reformulating a differential equation into the Sturm-Liouville form. In this exercise, these functions are derived directly from comparing the terms of the given differential equation with the standard Sturm-Liouville form.

• \( p(x) \): This is the function that precedes the second derivative term. In our example, the expression \( \frac{d}{dx}\left[x^2 \frac{dy}{dx}\right] \) allows us to identify \( p(x) = x^2 \).
• \( q(x) \): This function is associated with the \( y \) term itself. By identifying \((x^2 - u^2)y\) as the term in question, we set \( q(x) = x^2 - u^2 \).
• \( r(x) \): This is typically a positive function included with \( \lambda y \). Since \( \lambda = 0 \) in our problem, \( r(x) \) can be any constant, hence it is taken as \( r(x) = 1 \) to keep things simple.

Understanding these identifications helps in re-writing the differential equation in a manner that is simpler to solve and analyze.