Bessel's equation - 3

[logistic_guy]

Solve.

$\displaystyle x^2y'' + xy' + \left(x^2 - \frac{9}{4}\right)y = 0$

 

[logistic_guy]

If you followed the first thread of Bessel's equation, you would say instantly that the general solution to this differential equation is:

$\displaystyle y(x) = c_1J_{3/2}(x) + c_2J_{-3/2}(x)$

You are right, but most of us are simple people and we still don't understand what the heck $\displaystyle J$ means! Therefore, you need to write the solution in terms of trigonometric functions or anything else that we can understand.

You are very lucky buddy as soon you will understand everything about $\displaystyle J$ or $\displaystyle J_n$. For now it is enough that you know that $\displaystyle J$ is called Bessel's function of the first kind and when we write it like this $\displaystyle J_n$ it is called Bessel's function of the first kind of order $\displaystyle n$.

Now I will give you a magical recurrence relation for Bessel functions that will help you a lot to solve tons of Bessel's equations. And I promise you that the proof of this relation will be given in another Episode. For now just mimic how to use it!

The recurrence relation is:

$\displaystyle 2nJ_n(x) = xJ_{n+1}(x) + xJ_{n-1}(x)$

Let us try to plug $\displaystyle n = \frac{1}{2}$ and see what we get.

$\displaystyle 2\frac{1}{2}J_{1/2}(x) = xJ_{1/2+1}(x) + xJ_{1/2-1}(x)$

$\displaystyle J_{1/2}(x) = xJ_{3/2}(x) + xJ_{-1/2}(x)$

This gives:

$\displaystyle J_{3/2}(x) = \frac{1}{x}J_{1/2}(x) - J_{-1/2}(x)$

From previous lessons, we know that:

$\displaystyle J_{1/2}(x) = \sqrt{\frac{2}{\pi x}}\sin x$

And

$\displaystyle J_{-1/2}(x) = \sqrt{\frac{2}{\pi x}}\cos x$

Then

$\displaystyle J_{3/2}(x) = \frac{1}{x}\sqrt{\frac{2}{\pi x}}\sin x - \sqrt{\frac{2}{\pi x}}\cos x$

If we do the same steps but with $\displaystyle n = -\frac{1}{2}$, we get:

$\displaystyle J_{-3/2}(x) = -\sqrt{\frac{2}{\pi x}}\sin x - \frac{1}{x}\sqrt{\frac{2}{\pi x}}\cos x$

Then, the general solution in terms of trigonometric functions is:

$\displaystyle y(x) = c_1\left(\frac{1}{x}\sqrt{\frac{2}{\pi x}}\sin x - \sqrt{\frac{2}{\pi x}}\cos x\right) + c_2\left(-\sqrt{\frac{2}{\pi x}}\sin x - \frac{1}{x}\sqrt{\frac{2}{\pi x}}\cos x\right)$

Or

$\displaystyle y(x) = \left(C_1\frac{\sin x}{x^{3/2}} + C_3\frac{\cos x}{\sqrt{x}}\right) + \left(C_2\frac{\sin x}{\sqrt{x}} + C_4\frac{\cos x}{x^{3/2}}\right)$

Or

$\displaystyle y(x) = C_1\frac{\sin x}{x^{3/2}} + C_2\frac{\sin x}{\sqrt{x}} + C_3\frac{\cos x}{\sqrt{x}} + C_4\frac{\cos x}{x^{3/2}}$


Welcome to the world of differential equations