Is there a name for this type of differential equation?

Question

So I'm dealing with this type of differential equation: $$x^2y^{\prime\prime}+xy^{\prime}-(k^2x^2+1)y=0$$ where k is a positive real number. Now, this looks similar to the Modified Bessel equation, but searching along those lines didn't help. So, does this equation have a name, and can it be solved?

Answer 1

For $\mathrm k=0$, the equation is Cauchy-Euler equation. For $\mathrm k\neq0$ let $\mathrm{u=kx}$.

$$\mathrm{ {dy\over dx}=k{dy\over du},\quad {d^2y\over dx^2}=k^2{d^2y\over du^2} }$$ So our equation becomes $$\mathrm{ u^2{d^2y\over du^2}+u{dy\over du}-(u^2+1)y=0 }$$ which is modified Bessel equation. Therefore the required solution is $$\mathrm{ y=c_1I_1\left({kx}\right)+c_2K_1\left({kx}\right) }$$

Generalization

The substitution $\mathrm{u={2\over n}\sqrt{b}x^{n\over2}}$ turns $$\mathrm{ x^2y''+xy'-(bx^n+c)y=0 }$$ into a modified Bessel equation where $\mathrm b>0$.