stochastic processes - demonstrate that a Square-root process is Non-central Chi-squared distributed

how can i prove that the value at some future time $t'$, $x_{t'}$, of the Square-root process at current time $t$, $x_t$, is Chi-squared distributed?

$dx_t = k(\theta - x_t)dt + \beta \sqrt{x_t}dz_t$

explicitely:

$x_{t'} = x_t e^{-k(t'-t)} + \theta (1 - e^{-k(t'-t)}) + \beta \int_t^{t'} e^{-k(t'-u)}\sqrt{x_u}dz_u$

I just got to mean and variance by Ito's isometry:

$E_t[x_{t'}] = \theta + (x_t - \theta) e^{-k(t'-t)}$

and

$Var_t[x_{t'}] = \frac{\beta^2 x_t}{k} (e^{-k(t'-t)} - e^{-2k(t'-t)}) + \frac{\beta^2 \theta}{2k} (1 - e^{-k(t'-t})^2$

in the Ornstein-Uhlenbeck case there's no $\sqrt{x_t}$ in the volatility and therefore the stochastic integral is Normally distributed and everything is fine

unfortunately, I've never met the Non-central Chi-squared distribution before, so I'm not able to understand how to get to it (also because it seems itself pretty a mess to me)