Question

Find the Wronskian of two solutions of the given differential equation without solving the equation. \(x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-v^{2}\right) y=0, \quad\) Bessel's equation

Short answer

Answer: The Wronskian of two solutions of the given Bessel's equation, \(J_v(x)\) and \(Y_v(x)\), is given by: $$ W(J_v, Y_v) = \frac{Y_v'(x)[(2x+1)J_v''(x)+(2x-v^2)J_v'(x)] \\- J_v'(x)[(2x+1)Y_v''(x)+(2x-v^2)Y_v'(x)]}{x^2} $$

Step-by-step solution

Differentiate the equation once

Let's differentiate the given Bessel's equation with respect to \(x\): $$ \frac{d}{dx}\left(x^2y''+xy'+(x^2-v^2)y\right) = 0 $$ Which yields: $$ 2xy''+x^2y'''+y'+2xy'+(2x-v^2)y' = 0 $$ Rearranging and simplifying, we get: $$ x^2y'''+(2x+1)y''+ (2x-v^2)y'=0 $$

Substitute the solutions' form

Now we substitute the general solutions of Bessel's equation \(J_v(x)\) and \(Y_v(x)\) and their derivatives in this equation: For \(J_v(x)\), $$ x^2J_v'''(x)+(2x+1)J_v''(x)+(2x-v^2)J_v'(x)=0 $$ And for \(Y_v(x)\), $$ x^2Y_v'''(x)+(2x+1)Y_v''(x)+(2x-v^2)Y_v'(x)=0 $$

Multiply the equations and calculate the determinant

Now we multiply the equation for \(J_v(x)\) by \(Y_v'(x)\) and subtract the equation for \(Y_v(x)\) multiplied by \(J_v'(x)\): $$ Y_v'(x)[x^2J_v'''(x)+(2x+1)J_v''(x)+(2x-v^2)J_v'(x)]\\- J_v'(x)[x^2Y_v'''(x)+(2x+1)Y_v''(x)+(2x-v^2)Y_v'(x)]=0 $$ The left-hand side of this equation is the Wronskian determinant: $$ W(J_v, Y_v) = J_v(x)Y_v'(x) - Y_v(x)J_v'(x) $$

Simplify and solve for the Wronskian

We divide the entire equation by \(x^2\) and simplify: $$ W(J_v, Y_v) = \frac{Y_v'(x)[(2x+1)J_v''(x)+(2x-v^2)J_v'(x)] \\- J_v'(x)[(2x+1)Y_v''(x)+(2x-v^2)Y_v'(x)]}{x^2} $$ Since we have found the Wronskian without solving the Bessel's equation explicitly, we have completed the task as required. The Wronskian of two solutions of the given Bessel's equation is: $$ W(J_v, Y_v) = \frac{Y_v'(x)[(2x+1)J_v''(x)+(2x-v^2)J_v'(x)] \\- J_v'(x)[(2x+1)Y_v''(x)+(2x-v^2)Y_v'(x)]}{x^2} $$

Key concepts

Differential Equations

Differential equations are mathematical equations that relate some function with its derivatives. In practice, these functions usually represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. They are central to many fields such as physics, engineering, and finance because they model the dynamic changes of systems over time.

In the context of the Wronskian for Bessel's equation, the differential equation at hand is a second-order linear homogeneous differential equation with variable coefficients. This type of equation is challenging because it is hard to solve using elementary methods, which is why the properties of the solutions, like the Wronskian, are studied without actually solving the equation.

Bessel Functions

Bessel functions, denoted as \(J_v(x)\) and \(Y_v(x)\), are solutions to Bessel's differential equation which appears in a wide variety of physical problems, including heat conduction, wave propagation, and static potentials. These functions are named after Friedrich Bessel, who studied them in the context of planetary motion. Bessel functions are particularly noted for their oscillatory behavior and are categorized into different types, with \(J_v(x)\) called the Bessel function of the first kind and \(Y_v(x)\) called the Bessel function of the second kind. They are defined for any real or complex number \(v\), known as the order of the Bessel function.

Wronskian Determinant

The Wronskian determinant is a function used in the study of differential equations, named after the Polish mathematician Józef Hoene-Wronski. It is defined for two functions that are solutions to a differential equation and provides information about their linear independence. If the Wronskian is zero at some point and the functions and their first derivatives are continuous, then it implies the functions are linearly dependent.

For Bessel's equation, the Wronskian of two solutions \(J_v(x)\) and \(Y_v(x)\), represented as \(W(J_v, Y_v)\), helps to identify that the solutions are linearly independent, without having to solve the equation explicitly. This is crucial because the theory of differential equations tells us that if we have two linearly independent solutions to a second-order linear differential equation, we can express the most general solution as a linear combination of these two solutions.