Matrix differential equation with inequality, $ y'(t) \leq A y(t) $

Question

I have an $n \times n$ matrix $A$ and a system of ODEs of the form $$ y'(t) = A y(t) + b(t) $$ Where $b(t) \leq 0$ such that $$ y'(t) \leq A y(t) \qquad (1)$$ I also know that $y(t) \geq 0$. Is is correct to conclude from $(1)$ that $ 0 \leq y(t) \leq c_{1}e^{\lambda_{1}}u_{1} + ... + c_{n}e^{\lambda_{n}}u_{n} $ where $\lambda_{i}$ and $u_{i}$ are respectively the corresponding eigenvalues and eigenvectors of matrix $A$? I know that this should hold with equalities instead of inequalities but I am not sure if it does in this case as well (assuming $A$ has $n$ linearly independent eigenvectors).