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)