Explicit solution to Bessel-like equations

Question

I was wondering if one can explicitly solve the following equation $$ x^2 y'' + x y' + k^2 x^2 (x^\beta+1) y = a^2 y $$ for real constants $k,a,\beta$? If $\beta = 0$ then the solution is a linear combination of Bessel $Y_a$ and $J_a$.

Answer 1

Except for $\beta=2$, I do not think that we could have explicit solutions.

For that specific case

$$y=2^{\frac{a+1}{2}} x^a e^{\frac{1}{2} i k x^2} \left(c_1\, U\left(\frac{2 a-i k+2}{4} ,a+1,-i k x^2\right)+c_2\, L_{-\frac{2 a-i k+2-2}{4} }^a\left(-i k x^2\right)\right)$$

Answer 2

Assume $k\neq0$ for the key cases

$x^2y''+xy'+k^2x^2(x^\beta+1)y=a^2y$

$x^2y''+xy'+(k^2x^2(x^\beta+1)-a)y=0$

Which is a more generalized version compare to Bessel-like differential equation

Some special cases:

$\beta=0,-2$ : convertible to Bessel ODE

$\beta=-1,2$ : convertible to degenerate hypergeometric ODE

$\beta=-4$ : similar to Hunt for exact solutions of second order ordinary differential equations with varying coefficients. and can reduce to the doubly-confluent Heun equation