Homework Help: Bessel Differential Equation Problem

1. Jun 1, 2012

Hiche

1. The problem statement, all variables and given/known data

Use the substitution $x = e^t$ to solve the following differential equation in terms
of Bessel functions:

$\frac{d^{2}y}{dt^2} + (e^{2t} - \frac{1}{4})y = 0$

2. Relevant equations

3. The attempt at a solution

So, using the Chain Rule, $\frac{d^{2}y}{dt^2} = e^{2t}\frac{d^{2}y}{dx^2} = x^2\frac{d^{2}y}{dx^2}$, so our differential equation becomes:$$x^2\frac{d^{2}y}{dx^2} + (x^{2} - \frac{1}{4})y = 0$$.
The general solution is $y = c_1J_{1/2}(x) + c_2J_{-1/2}(x)$. After that we need to replace x with $e^t$. Is this correct?

The second question asks to express our answer in terms of the elementary functions. What is exactly meant by this?

2. Jun 1, 2012

tt2348

Actually, using the chain rule would give (y(x))'=y'(x)*x'(t)
a second application (y(x))''=(y'(x)*x'(t))=y''(x)*(x'(t))^2)+y'(x)*x''(t)

3. Jun 1, 2012

tt2348

elementary functions means answer it in terms of regular functions, hint sine and cosine (if this were a linear homogeneous equation, what would it look like.)

4. Jun 1, 2012

HallsofIvy

In general, the Bessel functions cannot be written in terms of elementary functions- which is why Bessel functions have whole books devoted to them! However, the Bessel functions of order 1/2 can be:
$$J_{1/2}(x)= \sqrt{\frac{2}{\pi x}}sin(x)$$
$$J_{-1/2}(x)= \sqrt{\frac{2}{\pi x}}cos(x)$$