(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data, relevant equation

[tex] x^{2}y'' + xy' + (4x^{4}-\frac{1}{4})y = 0[/tex]

2. The attempt at a solution

I tried substituting z = x^{2}

From this I have [tex]\frac{dy}{dx} = 2x \frac{dy}{dz}[/tex]

and [tex]\frac{d}{dx}(2x\frac{dy}{dz}) = 2xy'' + 2y'[/tex]

Then the original equation becomes:

[tex]2z^{3/2}y'' + 4zy' + (4z^{2}-\frac{1}{4})y = 0[/tex]

where derivatives of y are now with respect to the new variable z.

This does not look like a Bessel equation and I'm not sure how to make it look like one. Did I use the wrong substitution?

I know how to solve once it's in the correct form, but could someone help me get it there please?

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# Solve Bessel's equation through certain substitutions

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