General solution of Bessel's equation

 Explain and express general solution of Bessel's equation

Bessel's equation is a second-order linear differential equation that appears in various physical contexts, such as heat conduction, wave propagation, and electromagnetism. The general form of Bessel's equation is:

$x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - n^2) y = 0$

where $n$ is a constant, which can be any real or complex number. The solutions to this equation are known as Bessel functions.

To derive the general solution of Bessel's equation, we start with the standard form of Bessel's equation of order 

$n$:

$x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - n^2) y = 0$

We'll use the method of Frobenius to find the solution, which involves looking for a solution in the form of a power series:

$y(x) = \sum_{k=0}^{\infty} a_k x^{k+r}$

where $a_k$ are the coefficients to be determined, and $r$ is the exponent to be determined from the indicial equation.

Step 1: Substitute the Power Series into the Differential Equation

First, we need the derivatives of $y(x)$:

$\frac{dy}{dx} = \sum_{k=0}^{\infty} a_k (k+r) x^{k+r-1}$ 

$\frac{d^2 y}{dx^2} = \sum_{k=0}^{\infty} a_k (k+r)(k+r-1) x^{k+r-2}$

Substituting these into the Bessel's equation gives:

$x^2 \sum_{k=0}^{\infty} a_k (k+r)(k+r-1) x^{k+r-2} + x \sum_{k=0}^{\infty} a_k (k+r) x^{k+r-1} + (x^2 - n^2) \sum_{k=0}^{\infty} a_k x^{k+r} = 0$

x2∑k=0∞​ak​(k+r)(k+r−1)xk+r−2+x∑k=0∞​ak​(k+r)xk+r−1+(x2−n2)∑k=0∞​ak​xk+r=0

Simplify each term:

$\sum_{k=0}^{\infty} a_k (k+r)(k+r-1) x^{k+r} + \sum_{k=0}^{\infty} a_k (k+r) x^{k+r} + \sum_{k=0}^{\infty} a_k x^{k+r+2} - n^2 \sum_{k=0}^{\infty} a_k x^{k+r} = 0$

∑k=0∞​ak​(k+r)(k+r−1)xk+r+∑k=0∞​ak​(k+r)xk+r+∑k=0∞​ak​xk+r+2−n2∑k=0∞​ak​xk+r=0

Step 2: Combine Like Terms

Combine the series:

$\sum_{k=0}^{\infty} a_k (k+r)(k+r-1) x^{k+r} + \sum_{k=0}^{\infty} a_k (k+r) x^{k+r} + \sum_{k=0}^{\infty} a_k x^{k+r+2} - n^2 \sum_{k=0}^{\infty} a_k x^{k+r} = 0$

∑k=0∞​ak​(k+r)(k+r−1)xk+r+∑k=0∞​ak​(k+r)xk+r+∑k=0∞​ak​xk+r+2−n2∑k=0∞​ak​xk+r=0

Rewrite the third sum to match the powers of $x$:

$\sum_{k=0}^{\infty} a_k (k+r)(k+r-1) x^{k+r} + \sum_{k=0}^{\infty} a_k (k+r) x^{k+r} + \sum_{k=2}^{\infty} a_{k-2} x^{k+r} - n^2 \sum_{k=0}^{\infty} a_k x^{k+r} = 0$

∑k=0∞​ak​(k+r)(k+r−1)xk+r+∑k=0∞​ak​(k+r)xk+r+∑k=2∞​ak−2​xk+r−n2∑k=0∞​ak​xk+r=0

Now we have:

$\sum_{k=0}^{\infty} \left[ a_k (k+r)(k+r-1) + a_k (k+r) + a_{k-2} - n^2 a_k \right] x^{k+r} = 0$

Step 3: Indicial Equation and Recurrence Relation

The indicial equation is obtained by setting $k = 0$:

$a_0 r(r-1) + a_0 r - n^2 a_0 = 0$

$r(r-1) + r - n^2 = 0$

$r^2 - n^2 = 0$

$r = \pm n$

So, the roots are $r_1 = n$ and $r_2 = -n$.

Next, we find the recurrence relation by setting the coefficients of $x^{k+r}$ to zero for $k \ge 2$:

$a_k \left[ (k+r)(k+r-1) + (k+r) - n^2 \right] + a_{k-2} = 0$

$a_k \left[ k^2 + 2kr + r^2 - n^2 \right] + a_{k-2} = 0$

$a_k \left[ (k+r)^2 - n^2 \right] + a_{k-2} = 0$

Step 4: Solving the Recurrence Relation

For $r = n$:

$a_k \left[ (k+n)^2 - n^2 \right] + a_{k-2} = 0$

$a_k \left[ k^2 + 2kn \right] + a_{k-2} = 0$

$a_k k(k+2n) + a_{k-2} = 0$

$a_k = -\frac{a_{k-2}}{k(k+2n)}$

For $r = -n$:

$a_k \left[ (k-n)^2 - n^2 \right] + a_{k-2} = 0$

$a_k \left[ k^2 - 2kn \right] + a_{k-2} = 0$

$a_k k(k-2n) + a_{k-2} = 0$

$a_k = -\frac{a_{k-2}}{k(k-2n)}$

Step 5: General Solution

The general solution to Bessel's equation is the linear combination of the two independent solutions $y_1(x)$ and $y_2(x)$ corresponding to $r = n$ and $r = -n$:

$y(x) = A J_n(x) + B J_{-n}(x)$

where $J_n(x)$ and $J_{-n}(x)$ are the Bessel functions of the first kind of order $n$ and $-n$, respectively. For non-integer $n$, $J_{-n}(x)$ is independent of $J_n(x)$. For integer $n$, $J_{-n}(x)$ is not independent, and the second linearly independent solution is given by the Bessel function of the second kind, $Y_n(x)$:

$y(x) = A J_n(x) + B Y_n(x)$

Thus, we have derived the general solution of Bessel's equation.