Question

The Bessel equation of order one is $$ x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-1\right) y=0 $$ (a) Show that \(x=0\) is a regular singular point; that the roots of the indicial equation are \(r_{1}=1\) and \(r_{2}=-1 ;\) and that one solution for \(x>0\) is $$ J_{1}(x)=\frac{x}{2} \sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2 n}}{(n+1) ! n ! 2^{2 n}} $$ Show that the series converges for all \(x .\) The function \(J_{1}\) is known as the Bessel function of the first kind of order one. (b) Show that it is impossible to determine a second solution of the form $$ x^{-1} \sum_{n=0}^{\infty} b_{n} x^{n}, \quad x>0 $$

Short answer

In this problem, we were given the Bessel equation of order one and asked to: (a) Show that \(x=0\) is a regular singular point, find the roots of the indicial equation \(r_1\) and \(r_2\), find one solution for \(x>0\), the Bessel function of the first kind of order one \(J_1(x)\), and show that the series converges for all \(x\). (b) Show that it is impossible to determine a second solution in the given form. We used the Frobenius method to analyze the equation and determined that \(x = 0\) is a regular singular point with roots \(r_1 = 1\) and \(r_2 = -1\). We then found the Bessel function of the first kind of order one, \(J_1(x)\), and demonstrated that the series for \(J_1(x)\) converges for all \(x\). Finally, we attempted to find a second solution in the given form and showed that it is not possible to derive a second linearly independent solution using this method.

Step-by-step solution

(a) Regular Singular Point and Roots of Indicial Equation

The given Bessel equation is \(x^2y'' + xy' + (x^2 - 1)y = 0\). To check whether \(x = 0\) is a regular singular point, we must see if the following limits exist: 1. \(\lim_{x \to 0} x^2(x^2y'' + xy' + (x^2 - 1)y) = \lim_{x \to 0} x^2y'' + x^3y' + x^4y\) 2. \(\lim_{x \to 0} x(x^2y'' + xy' + (x^2 - 1)y) = \lim_{x \to 0} x^3y'' + x^2y' + xy\) The limits clearly exist for any finite functions \(y''\), \(y'\), and \(y\), which implies that \(x = 0\) is a regular singular point. To find the roots of the indicial equation, we need to apply the Frobenius method. Assume a solution of the form \(y(x) = x^r \sum_{n=0}^{\infty} a_n x^n\). By calculating the derivatives and substituting them into the Bessel equation, equating the coefficients of \(x^{r + n}\) to zero, we obtain the indicial equation: \(r(r - 1) + r = r^2 - 1 = 0\) The roots of this equation are \(r_1 = 1\) and \(r_2 = -1\).

(a) Bessel Function of the First Kind of Order One

Now, we need to find the solution \(J_1(x)\) when \(x>0\). We can use the root \(r_1=1\) and the Frobenius method, leading to: \(J_1(x) = x^1 \sum_{n=0}^{\infty} a_n x^n = x \sum_{n=0}^{\infty} a_n x^n\) By substituting \(J_1(x)\) and its derivatives into the Bessel equation and equating the coefficients of \(x^{n+1}\) to zero, we get the recurrence relation: \(a_{n+1} = \frac{(-1)^n}{(n+1)!(n+1)! (2^{2n})}a_0\) Let \(a_0 = \frac{1}{2}\) (we can set \(a_0\) as any constant, it still generates a valid solution). We can now substitute this equation back into our expression for \(J_1(x)\): \(J_1(x) = x \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(n+1) ! n! 2^{2n}}\) The series converges for all \(x\) by the ratio test, as the ratio of absolute values of consecutive terms is: \(R = \frac{(2n)!}{(n+1)!(n+1)!}\) Which converges to 0 as \(n\) goes to infinity, ensuring the convergence of the series.

(b) Proving It's Impossible to Find Second Solution in Given Form

Now we must show that we cannot determine a second solution of the form: \(S(x) = x^{-1} \sum_{n=0}^{\infty} b_n x^n\) We know that the solution of the Bessel equation must be linearly independent, which means that if \(S(x)\) is a solution, it must not be in the form of a constant times \(J_1(x)\). We start by considering the form of the second solution: \(S(x) = x^{-1} \sum_{n=0}^{\infty} b_n x^n = \sum_{n=0}^{\infty} b_n x^{n-1}\) Taking the first and second derivatives of \(S(x)\), we have: \(S'(x) = \sum_{n=0}^{\infty} b_n(n-1) x^{n-2}\) \(S''(x) = \sum_{n=0}^{\infty} b_n(n-1)(n-2) x^{n-3}\) Substituting these derivatives and \(S(x)\) into the Bessel equation gives us: \(\sum_{n=0}^{\infty} b_n(n-1)(n-2)x^{n} +\sum_{n=0}^{\infty} b_n(n-1)x^{n}+\sum_{n=0}^{\infty} b_n(x^{n+2}-x^{n})=0\) We must now compare the coefficients of \(x^n\), \(x^{n-1}\), and \(x^{n-2}\) in this equation. First, we observe that the coefficients of \(x^n\) yield the following recurrence relation: \(b_n(n-1)(n-2) + b_{n-1}(n-1) = 0\) However, this recurrence relation does not allow us to compute \(b_n\) for all \(n\). Specifically, it does not give us the expression for \(b_0\) or \(b_1\). This shows that it's impossible to determine a second linearly independent solution of the given form.

Key concepts

Understanding Regular Singular Points

In the study of differential equations, particularly Bessel's equation, the concept of a regular singular point is fundamental.

Consider a second-order linear differential equation. A point, say x = x0, is called a 'singular point' if at least one of the coefficients of the highest derivatives becomes infinite there. However, not all singular points behave the same. If certain limits involving the equation's coefficients at x0 exist, then that singular point is considered 'regular'.

The importance of determining whether a singular point is regular cannot be overstated, as the nature of the point dictates the method we use to find solutions. For the Bessel equation given in the exercise, when x equals zero, we indeed have a singular point. To qualify it as a 'regular' singular point, we confirm that the limits required to satisfy this classification exist. This knowledge gives us the go-ahead to apply specialized techniques, such as the Frobenius method, designed to handle such points effectively.

Deciphering the Indicial Equation

The indicial equation is critical when solving differential equations with regular singular points with the Frobenius method. It essentially determines the possible leading powers, or indices, in the series solutions to the differential equation.

Approaching a differential equation like Bessel's, once we've established the presence of a regular singular point, we look for solutions that are power series multiplied by some power of x. This is where the indicial equation comes into play. By substituting our assumed power series solution into the equation, we usually encounter a pattern in the coefficients, which can be distilled into what we call the indicial equation.

In our exercise, we are left with a rather simple indicial equation, which indicates the initial powers of x in our series solution. The roots of the indicial equation, in this case, are +1 and -1. These roots are crucial, as they inform us about the nature of possible solutions to the Bessel's equation and ensure we're considering the correct form of the solution.

Exploring the Bessel Function of the First Kind

A Bessel function of the first kind of order one, denoted as J_1(x), is a particular solution to Bessel's equation. The function serves many purposes in mathematical physics, such as describing wave propagation or heat conduction in specific symmetrical situations.

The derivation of J_1(x) involves leveraging the roots of the indicial equation obtained earlier. We then construct a power series solution with coefficients that satisfy the recurrence relation, resulting from substituting this series back into the differential equation.

Various properties of the Bessel function, like its convergence for all x, as shown through the ratio test in the given exercise, are vital for its applicability. It's fascinating to note that the function, despite being defined by an infinite series, converges absolutely and uniformly on any bounded interval. Consequently, the Bessel function of the first kind is an example of a well-behaved function that appears often in physics and engineering problems.