I want to obtain a solution for the ODE:
$$x^2 y''+xy'+(16x^2-\alpha^2)y=0$$
and I can see that it looks similar to a Bessel Equation:
$$x^2 y'' +xy'+(x^2-n^2)y=0$$
but I'm not sure how to "transform it"...
2026-03-26 12:16:52.1774527412
How to find the solution to this ODE in terms of Bessel Functions
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As @Kenny pointed out in his comment for the equation $$x^2 y''+xy'+(\beta^2x^2-\alpha^2)y=0$$ Simply substitute $z=\beta x$ the equation becomes a Bessel equation $$z^2 y''+zy'+(z^2-\alpha^2)y=0$$ With solution $$y(z)=C_1J_{\alpha}(z)+C_2Y_{\alpha}(z)$$
$$ \implies y(x)=C_1J_{\alpha}(\beta x)+C_2Y_{\alpha}(\beta x)$$