Solved: (Elementary Bessel functions) Derive (25) in
Chapter 5, Problem 5.1.111(choose chapter or problem)
Using the indicated substitutions, find a general solution in terms of \(J_{\nu}\) and \(J_{-\nu}\) or indicate when this is not possible. (This is just a sample of various ODEs reducible to Bessel's equation. Some more follow in the next problem set. Show the details of your work.)
\(x^{2} y^{\prime \prime}+x y^{\prime}+4\left(x^{4}-\nu^{2}\right) y=0 \quad\left(x^{2}=z\right)\)
Text Transcription:
J_nu
J_- nu
x^2 y “ + xy ‘ + 4(x^4 - nu^2) y = 0 (x^2 = z)
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