Question

Find a fundamental set of Frobenius solutions of Bessel's equation $$ x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-\nu^{2}\right) y=0 $$ in the case where \(\nu\) is a positive integer.

Short answer

Answer: The fundamental set of Frobenius solutions for the integer-order Bessel's equation is given by $$y(x)=\sum_{n=0}^{\infty} a_n x^{n+r},$$ where \(r=0,1,2,...,\nu-1, \nu+1,...\) and \(a_n\) can be found using the recurrence relation \((n+r)(n+r-1)+(n+r)-\nu^2=0.\) This is derived by applying the Frobenius method, which involves assuming a solution of the form: $$y(x) = x^{r} \sum_{n=0}^{\infty} a_nx^n,$$ calculating the first and second derivatives, substituting these derivatives into the given Bessel's equation, rewriting the equation as a single series, and equating the coefficients of like powers of x to form a recurrence relation for \(a_n\).

Step-by-step solution

Assume a Frobenius solution

Assume a solution of the form: $$y(x) = x^{r} \sum_{n=0}^{\infty} a_nx^n.$$ Where \(r\) is an indeterminate exponent.

Calculate derivatives

Calculate the first and second derivatives of the assumed solution: $$y'(x)=\sum_{n=0}^{\infty} (n+r)a_nx^{n+r-1},$$ $$y''(x)=\sum_{n=0}^{\infty}(n+r)(n+r-1)a_nx^{n+r-2}.$$

Substitute into the Bessel's equation

Substitute these derivatives into the given Bessel's equation and simplify: $$x^2\sum_{n=0}^{\infty}(n+r)(n+r-1)a_nx^{n+r-2} + x\sum_{n=0}^{\infty} (n+r)a_nx^{n+r-1} + \left(x^2-\nu^2\right)\sum_{n=0}^{\infty} a_nx^{n+r} = 0.$$

Rewrite the equation as a series

Rewrite the equation as a single series with the same power of x: $$\sum_{n=0}^{\infty}\left[(n+r)(n+r-1)a_nx^{n+r}+(n+r)a_nx^{n+r}-\nu^2a_nx^{n+r}\right]=0.$$

Form a recurrence relation

Since the series must equal zero, the coefficients of like powers of x must also be zero. Therefore, we can equate the coefficients to zero: \((n+r)(n+r-1)a_n+(n+r)a_n-\nu^2a_n=0.\) Because \(a_n\neq0\), we can divide the entire equation by \(a_n\): \(R_n = (n+r)(n+r-1)+(n+r)-\nu^2=0,\) where \(R_n\) is the indeterminate in the sequence.

Solve the indeterminate equation

Solve the indeterminate equation: \(r=0,1,2,...,\nu-1, \nu+1,...\)

Write the fundamental set of Frobenius solutions

Using the solutions for \(r\) found in the previous step, we can now write the fundamental set of Frobenius solutions for the Bessel's equation: $$y(x)=\sum_{n=0}^{\infty} a_n x^{n+r},$$ where \(r=0,1,2,...,\nu-1, \nu+1,...\) and \(a_n\) can be found using the recurrence relation given in step 5.

Key concepts

Bessel's equation

Bessel's equation is a differential equation that appears frequently in various fields of physics and engineering. The standard form of Bessel's equation is given by: \[ x^{2} y^{\prime \prime} + x y^{\prime} + \, (x^{2} - u^{2}) y = 0. \] This equation arises when solving problems involving cylindrical symmetry, such as heat conduction in a cylindrical object or vibration modes of a circular membrane. Here, \( u \) is a parameter known as the order of the Bessel function, which typically takes on non-negative integer values in practical applications. The challenge with Bessel's equation is that it cannot usually be solved using simple elementary functions. Instead, solutions to this equation are special functions known as Bessel functions, denoted as \( J_u(x) \). These functions play a crucial role in describing wave behavior and other physical phenomena.

recurrence relation

A recurrence relation is an equation that relates terms in a sequence or array. In the context of solving differential equations like Bessel's equation, recurrence relations are used to find coefficients in a series solution. Such relations help express each coefficient in terms of previous coefficients, facilitating computation of solutions. When applying the Frobenius method to Bessel's equation, a solution is found as a power series: \[ y(x) = x^{r} \sum_{n=0}^{\infty} a_n x^n. \] Substituting the series into Bessel's equation results in a recurrence relation among the coefficients \( a_n \). Simplifying the terms, every coefficient needs to satisfy: \[ (n+r)(n+r-1)a_n + (n+r)a_n - u^2 a_n = 0. \] By setting these equal to zero, recursive values of \( a_n \) are determined. This allows for the generating of terms in the power series solution, providing insight into the solution's structure.

indeterminate exponent

In the Frobenius method, the indeterminate exponent is a parameter of crucial significance in expanding solutions as a power series around a singular point. When dealing with differential equations like Bessel's equation, the indeterminate exponent, \( r \), is a part of the initial series solution: \[ y(x) = x^{r} \sum_{n=0}^{\infty} a_n x^n. \] Finding the correct value of \( r \) is vital because it influences the convergence and structure of the solution. Indeterminate, because it is not known at the beginning of the solution process, \( r \) is determined by solving an indicial equation, derived from setting the lowest power of \( x \) in the series to zero. The determination of \( r \) allows identification of the structure and potential patterns of the solution, serving as a foundational step in the Frobenius method.

fundamental set of solutions

The fundamental set of solutions refers to a pair of linearly independent solutions to a differential equation, sufficient to construct the general solution. For Bessel's equation, a fundamental set of Frobenius solutions is used to represent all possible solutions of the equation around a regular singular point. Given Bessel's equation with integer \( u \), the solutions take the form: \[ y(x) = \sum_{n=0}^{\infty} a_n x^{n+r}, \] where \( r \) values correspond to specific roots of the indicial equation. These roots are crucial as they guide the construction of the fundamental set. In scenarios where \( u \) is a positive integer, these roots include both zero and positive integers excluding \( u \). This provides two linearly independent solutions needed for the complete solution set, encompassing all behavior of the original problem.