Solutions for a a nonlinear system expressed in state space form

Question

If for a linear system in state-space representation given by $$\begin{cases} \dot x(t) =A(t)x(t)+B(t)u(t)\\ y(t) =C(t)x(t)+D(t)u(t) \end{cases}$$ the state and output solutions are given by $$\begin{cases} x(t)=e^{A(t-t_0)}x(t_0)+\int_{t_0}^{t}e^{A(t-\tau)}Bu(\tau)d\tau\\ y(t)=Ce^{A(t-t_0)}x(t_0)+C\int_{t_0}^{t}e^{A(t-\tau)}Bu(\tau)d\tau+Du(t), \end{cases}$$ In case of a nonlinear system given in state space representation as $$\begin{cases}\dot x=f(t,x,u)\\ y=g(t,x,u)\end{cases}$$ there is a general procedure to find out the corresponding solutions? Or I have always to apply a linearization to boil down to a linear representation? Or alternatively we have to proceede with numerical methods?