I'm pretty sure the Bessel function $J_0(x)$ should solve the equation:
$$ x^2 \frac{d^2 f}{d x^2} + x \frac{d f}{dx} + x^2 f = 0 $$
according to wikipedia this is the standard form of a bessel function differential equation. However, when I go to check this solution, using the identities:
$$ \frac{d}{dz} J_0(x) = - J_1(x) $$ and $$ \frac{d}{dz} J_1(x) = J_0(x) - \frac{1}{2} J_1(x) $$
I get
$$ x^2 \frac{d^2 J_0}{dx^2 } + x \frac{d J_0}{dx} + x^2 J_0 = 0 $$
$$ -x^2 \frac{d }{dx } J_1(x) - x J_1(x) + x^2 J_0 = 0 $$
$$ -x^2 \left ( J_0(x) - \frac{1}{2} J_1(x) \right ) - x J_1(x) + x^2 J_0 = 0 $$
$$\left ( \frac{x}{2} - 1 \right ) x J_1(x) = 0 $$
However, this does not seem to be true, where am I going wrong?