Is $\sqrt{x} J_0(x)$ solution to any second-order differential equation?

Question

The Bessel function of the first kind $J_0(x)$ is a solution to the differential equation

$$ x^2 \frac{\mathrm{d}^2 f}{\mathrm{d}x^2} + x \frac{\mathrm{d}f}{\mathrm{d}x} + x^2f = 0.$$

To what second-order differential equation (if any) is $\sqrt{x} J_0(x)$ a solution to?

Answer 1

Taking the advice of @Cameron Williams, we can simply define $g(x) = \sqrt{x}J_0(x)$ and substitute $f(x) = \frac{g(x)}{\sqrt{x}}$ into the Bessel equation, which yields

$$4x^2 \frac{\mathrm{d}^2g}{\mathrm{d}x^2} + (1+4x^2)g = 0$$

Answer 2

Hint: try calculating $$4x^2\frac{d^2f}{dx^2}+f$$