I am currently studying stochastic differential equations and have no glue how to prove existence of a weak solution. For instance, consider $$ \begin{cases} dX_t=dB_t+\sqrt{X_t}dt\\ X_0=0 \end{cases} $$ on a compact interval $[0,T]$.
Any help is appreciated.
Welll, if your only concern is existence, then there are some results that you can use. For $$ X_t = x + \int_0^t D(X_s) dB_s + \int_0^t C(X_s)ds, t \in \mathbb{R^+}, x \in \mathbb{R}^d $$ where $C,D$ are measurbale, then a weak solution exists if $C$ and $D$ satisfies $$ \begin{align*} &|C(x)|^2 + ||D \circ D^{*}(x)|| \le K(1+|x|^2), x \in \mathbb{R}^d \\ &C \circ C^{*}(x) \ge \varepsilon I, x \in \mathbb{R}^d \end{align*} $$ for some $K, \varepsilon > 0$ where $I$ is the $d \times d$ identity matrix. For a reference see On existence and stability of weak solutions of multidimensional stochastic differential equations with measurable coefficients by Rozkosz and Slominski.
In your case, it will reduce to check if $ 1 + |x| \le K(1+x^2)$ for some K and $1 \ge \varepsilon \cdot 1$ for some $\varepsilon >0$. These conditions can easily be verified, so a weak solution should exist.