Non Homogeneous Linear Systems

$$ \begin{align*} \text{DE: } \frac{dX}{dt}&= A(t)X+F(x)\\ \text{A solution: }\phi_{0}(t)&= \phi(t)\int_{t_{0}}^{t}\phi^{-1}(u)F(u)du\\ \text{where fundamental matrix of corresponding homogeneous: }\phi(t)\\\ \text{General solution satisfying BVP }X(t_{0})=x_{0}\\ X(t)&= \phi(t)\phi^{-1}(t_{0})x_{0}+ \phi_{0}(t)\\ \end{align*} $$