Question

The differential equation \(t^{2} y^{\prime \prime}+t y^{\prime}-\left(t^{2}+v^{2}\right) y=0\) is known as the modified Bessel equation. Its solutions, usually denoted by \(I_{v}(t)\) and \(K_{v}(t)\), are called modified Bessel functions. This equation arises in solving certain partial differential equations involving cylindrical coordinates. (a) Do you anticipate that the modified Bessel equation will possess solutions that are even and odd functions of \(t\) ? Explain. (b) The point \(t=0\) is a regular singular point of the modified Bessel equation; all other points are ordinary points. Determine the indicial equation for the singularity at \(t=0\) and find the exponents at the singularity. (c) Obtain the recurrence relation for the modified Bessel equation. How do the exponents and recurrence relation for this equation compare with their counterparts

Short answer

#Answer# (a) The modified Bessel equation could possess even and odd solutions. (b) The exponents at the singularity are \(r = \frac{1 \pm \sqrt{1+4v^2}}{2}\). (c) The recurrence relation for the modified Bessel equation is \((n+2)(n+2r+1)a_{n+2} = (v^2 - n)a_n\). This differs from the Bessel equation, where the exponents are \(r = \pm v\) and the recurrence relation is \((n+2)(n+2r)a_{n+2} = (1 - n^2 - v^2)a_n\).

Step-by-step solution

Part (a) - Even and Odd Solutions

Even and odd functions satisfy the properties: \(f(-t) = f(t)\) for even functions and \(f(-t) = -f(t)\) for odd functions. The Bessel equation is: \(t^{2} y^{\prime \prime}+t y^{\prime}-\left(t^{2}+v^{2}\right) y = 0\) Let's consider \(y(-t)\): \((-t)^{2} y^{\prime \prime}(-t)+(-t) y^{\prime}(-t)-\left((-t)^{2}+v^{2}\right) y(-t) = t^{2} y^{\prime \prime}(-t)-t y^{\prime}(-t)-\left(t^{2}+v^{2}\right) y(-t) = 0\) The above equation is equivalent to the original equation. Therefore, the modified Bessel equation could possess even and odd solutions.

Part (b) - Indicial Equation and Exponents at Singularity

Using the Frobenius method, we assume a solution of the form \(y(t) = \sum_{n=0}^{\infty} a_n t^{n+r}\), where \(r\) is an unknown exponent. Plugging this into the Bessel equation, and equating the powers of \(t\), we can find the indicial equation. First, we calculate the derivatives: \(y'(t) = \sum_{n=0}^{\infty} (n+r)a_n t^{n+r-1}\) \(y''(t) = \sum_{n=0}^{\infty} (n+r)(n+r-1)a_n t^{n+r-2}\) Substituting these into the equation, we get: \(\sum_{n=0}^{\infty} \left( (n+r)(n+r-1)a_n t^{n+r} + (n+r)a_n t^{n+r} - (t^{2}+v^{2})a_n t^{n+r} \right) = 0\) The lowest power of \(t\) is \(t^r\). Equating the coefficients of \(t^r\) to zero, we obtain the indicial equation: \(r(r-1) + r - v^2 = 0\) Solving for \(r\), we find the exponents at the singularity: \(r = \frac{1 \pm \sqrt{1+4v^2}}{2}\)

Part (c) - Recurrence Relation and Comparison

To find the recurrence relation, we look at the coefficients of the rest of the powers of \(t\). By comparing the coefficients of equal powers in our Frobenius representation of the solution and the Bessel equation, we arrive at the recurrence relation: \((n+2)(n+2r+1)a_{n+2} = (v^2 - n)a_n\) For the modified Bessel equation, the exponents are \(r = \frac{1 \pm \sqrt{1+4v^2}}{2}\), and the recurrence relation is given by \((n+2)(n+2r+1)a_{n+2} = (v^2 - n)a_n\). On the other hand, for the Bessel equation, the exponents are \(r = \pm v\), and the recurrence relation is given by \((n+2)(n+2r)a_{n+2} = (1 - n^2 - v^2)a_n\). In summary, the exponents and recurrence relation for the modified Bessel equation differ from those of the Bessel equation. Specifically, the modified Bessel equation has a different exponent structure and a different coefficient in the recurrence relation.

Key concepts

Frobenius method

The Frobenius method is a powerful technique used to find solutions to linear differential equations around a singular point. It's especially useful when dealing with equations where regular power series don't easily apply, such as the case with modified Bessel functions.

In the exercise, we use the Frobenius method to tackle the modified Bessel equation. This equation, given by \(t^{2} y'' + t y' - (t^{2} + v^{2}) y = 0\), has a regular singular point at \(t = 0\). The beauty of the Frobenius method is that it lets us find solutions by assuming a series representation:

• We express the solution as \(y(t) = \sum_{n=0}^{\infty} a_n t^{n+r}\), where \(r\) is an unknown "indicial" exponent.
• This choice accommodates the behavior near the singularity by allowing both integer and non-integer powers of \(t\).

By substituting this series into the differential equation, we can generate a set of equations for the coefficients \(a_n\), known as the recurrence relation. This approach provides us with a systematic pathway to solve and understand complex differential equations.

Indicial equation

The indicial equation is crucial in determining the behavior of a differential equation at its singular points. For the modified Bessel equation, the point \(t = 0\) is identified as a regular singular point. This means the solution can still be expressed as a power series, but the leading exponent may not be an integer.

The indicial equation helps us find this leading exponent(s). Here's how we arrived at it in the exercise:

• We substitute the series \(y(t) = \sum_{n=0}^{\infty} a_n t^{n+r}\) into the differential equation.
• After combining like terms, we equate the coefficients of the lowest power of \(t\) (which is \(t^r\)) to zero. This yields the indicial equation.

For the modified Bessel equation, the indicial equation is \(r(r-1) + r - v^2 = 0\). Solving this quadratic equation gives us the possible values of \(r\), or exponents, which are \(\frac{1 \pm \sqrt{1+4v^2}}{2}\). These exponents characterize the behavior of the solution near \(t = 0\).

Recurrence relation

Once the indicial equation is solved, and potential exponents \(r\) are determined, the next step is to find the recurrence relation for the coefficients \(a_n\). This step in the Frobenius method ensures the series represents a valid solution to the differential equation.

The recurrence relation is derived by continuing to equate coefficients of like powers of \(t\) in the expanded form of the modified Bessel equation. In our exercise, we used the forms:

• Substitute the series solutions into the differential equation.
• Ensure all coefficients balance out for each power of \(t\).

For the modified Bessel equation, the relation we derive is: \((n+2)(n+2r+1)a_{n+2} = (v^2 - n)a_n\).

This equation allows us to recursively compute the coefficients \(a_n\) based on previous coefficients, creating a chain that, when solved, reconstructs the full series solution for the differential equation.

Singular points

Singular points of a differential equation, particularly regular singular points, are locations where the equation's behavior significantly changes. A regular singular point, such as \(t = 0\) in the modified Bessel equation, permits a series solution through the Frobenius method.

Understanding these points is essential for determining how to tackle the solution process and what form the solution might take. When comparing regular and irregular singular points:

• Regular singular points allow for Frobenius series solutions, offering a tidy approach to tackling potentially complex equations.
• Irregular singular points often require more advanced methods and can introduce complications like multi-valued or logarithmic solutions.

For the modified Bessel function in this exercise, each point except \(t = 0\) was ordinary, meaning straightforward solutions akin to power series could be found.

Recognizing singular points, particularly regular ones, frames how we proceed with finding solutions to differential equations, ensuring that our solution methods align with the nature of the equation.