$\text{Consider a gaussian sphere of radius r}$

$\text{Applying gauss law on this gaussian sphere}$

$\Rightarrow \int{\vec{E}.\vec{ds}} = \frac{q_{in}}{\epsilon_0}$

$\Rightarrow E.4\pi r^2 = \frac{1}{\epsilon_0} \int_{0}^{r}{\rho 4\pi r^2 dr}$

$\Rightarrow E = \frac{\rho_0}{\epsilon_0}(\frac{r}{3} - \frac{r^2}{4R})$