Bessel Differential Equation Problem

[Hiche]

Homework Statement

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

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

Homework Equations

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?

 

[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)

 

[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.)

 

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