Suppose there is a differential equation in the form of
$$\mathbf y'(t)=\begin{bmatrix} y_1+y_1^2 \\ y_2+y_2^2 \\ y_3+y_3^2 \end{bmatrix}.$$ What is the general method of finding the solution to such an equation? For a vector of purely linear terms, as in $$\mathbf y'(t)=\begin{bmatrix} 1 & 0 & 0 \\ 0 &1 & 0\\ 0 &0 &1 \end{bmatrix} \begin{bmatrix} y_1 \\ y_2 \\ y_3 \end{bmatrix},$$ the general solution is $$\mathbf y(t)=e^{At} \mathbf x_{0},$$ but I am not sure what the general solution or method is for a vector with non-linear components.
Or, suppose an even simpler case: $$\mathbf y'(t)=\begin{bmatrix} y_1+y_1^2 \\ y_2\\ \end{bmatrix}.$$