Question

Find all singular points of the given equation and determine whether each one is regular or irregular. \(x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-v^{2}\right) y=0, \quad\) Bessel equation

Short answer

Based on the given Bessel equation \(x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-v^{2}\right) y=0\), we found a singular point at \(x=0\). After checking the analyticity of the expressions \(P(x) = \frac{1}{x}\) and \(Q(x) = \frac{-v^2}{x^2}\), we concluded that the singular point at \(x=0\) is not regular. Therefore, the Bessel equation has an irregular singular point at \(x=0\).

Step-by-step solution

Identify the Singular Points

In order to find the singular points, we need to find when the coefficients of the leading terms become 0. In this case, we need to find the values of \(x\) when \(x^2\) and \(x\) become 0. Since both coefficients vanish when \(x=0\), we know that there is a singular point at \(x=0\).

Check if the Singular Point is Regular

To check if a singular point \(x_0\) is regular, we need to verify if the following expressions are analytic in some neighborhood of \(x_0\): \(P(x) = \frac{1}{x}\) \(Q(x) = \frac{-v^2}{x^2}\) We will check the analyticity of these expressions at \(x=0\). P(x) is not analytic at x = 0, as the function has a singularity at x = 0. Q(x) is also not analytic at x = 0, due to the division by \(x^2\). Hence, no analytic solutions can be found in a neighborhood of x = 0.

Classify the Singular Points

We have only one singular point for the given Bessel equation, which is \(x=0\). Based on the analysis in step 2, the point \(x=0\) is not regular, and therefore, it is an irregular singular point. Final classification of singular points: - Irregular singular point: x = 0

Key concepts

Bessel equation

The Bessel equation is a fundamental linear second-order differential equation with many applications in physics and engineering, particularly when problems involve cylindrical or spherical symmetry. The standard form of this equation is written as:
\[x^{2} y^{\text{''}}+x y^{\text{'}}+(x^{2}-v^{2}) y=0,\]
where \( y^{\text{''}} \) and \( y^{\text{'}} \) represent the second and first derivatives of \( y \) with respect to \( x \), respectively, and \( v \) is a constant that often relates to the order of the Bessel function, essentially determining its type.
This equation leads to solutions known as Bessel functions, which are essential in solving problems with circular or cylindrical boundaries. In the context of singular points, the Bessel equation shows its peculiarities at \( x=0 \) which is of significant interest when discussing singular points.

Regular singular point

In differential equations, particularly those with variable coefficients, a regular singular point is a type of singularity that, despite being a point where the equation's coefficients may be undefined or infinite, still allows the equation to have an analytically well-defined solution in the neighborhood of the point. This means that although the coefficients of the highest derivatives vanish or become infinite, the solutions can still be written in terms of a power series.
To determine if a singular point, denoted as \( x_0 \), is regular, the functions \( P(x) = \frac{1}{x-x_0} \) and \( Q(x) = \frac{a_0}{(x-x_0)^2} \), where \( a_0 \) is the coefficient of \( y \) in the equation, must both be analytic at \( x_0 \). If they are, then \( x_0 \) is a regular singular point; otherwise, it is an irregular singular point. Regular singular points are crucial as they imply the possibility of finding series solutions known as Frobenius series.

Irregular singular point

Opposed to a regular singular point, an irregular singular point is where a differential equation has coefficients that become infinite or cause discontinuity, and as a result, the solution to the differential equation cannot be expressed in terms of a convergent power series around that point. It is typically more challenging to find solutions near an irregular singular point because standard techniques like the power series expansion fail.
In the context of the Bessel equation, as indicated in the exercise solution, the point \( x=0 \) proves to be an irregular singular point because the functions \( P(x) \) and \( Q(x) \) are not analytic at this point. This irregularity is important in many physical situations and calls for alternative methods, such as asymptotic expansions or numerical approaches, to explore the behavior of the solutions.

Analyticity

Analyticity refers to the property of a function to be represented by a convergent power series around a point within its domain. This concept is pivotal in understanding the local behavior of solutions to differential equations. An analytic function can be differentiated any number of times, and its Taylor series about a point will converge to the function's value within some radius of convergence. In simpler words, if a function is analytic at a point, the function is well-behaved and smooth near that point.
When analyzing differential equations, checking for analyticity of the coefficients after transforming them into the function forms like \( P(x) \) and \( Q(x) \) helps determine the nature of singular points. If the transformed coefficients are analytic at the singular point, the solutions can often be expressed as an easily managable series. In the case of the exercise problem, the lack of analyticity of the Bessel equation coefficients at \( x=0 \) rendered the singular point as irregular.