Existence of solution for linear system of ODE Consider linear system of ODE given by
\begin{eqnarray}
u'_i(t)&=&\sum\limits_{i=1}^{n}a_{ij}(t)u_j+b_i(t) &\quad i=1,2,\ldots,n\\
u_i(0)&=&u_i^0 &\quad i=1,2,\ldots,n
\end{eqnarray}
Suppose $a_{ij},b_i \in L^{\infty}(\mathbb{R}^+)$ and $u_i^0\in \mathbb{R}$ for $i=1,2,\ldots,n,$ then do we have the existence of solution? ?If so how to prove it?
In other words, can we relax continuity assumption  on $a_{ij},b_i$ in the existence proof?

As the other answers demonstrate, there are obstacles to solving $u'=Au+b$ as differential equation with differentiable solutions. However, the associated integral equation
$$
u(t)=u(0)+\int_0^t(A(s)u(s)+b(s))\,ds
$$
is well-defined as fixed-point equation over continuous functions. As $A$ is essentially bounded, $L={\rm ess}\sup_t\|A(t)\|<\infty$ and
$$
\|u\|_L=\sup_t e^{-2L|t|}\|u(t)\|
$$
is a norm on a closed subspace of continuous functions that makes the fixed-point iteration to the above equation contracting with factor $\frac12$. Thus by the fixed-point theorem there is a unique solution.
