Is there an explicit ODE solution for this system?

Does this ODE system have solution?

• Let $A(t)$ and $B(t)$ be matrices with each element in $L^\infty(0,T).$ Let $A(t)$ have an inverse. I know nothing else about this inverse. Let $c(t)$ be a vector in $L^2(0,T).$ Let $u(t) \in \mathbb{R}^n$ be unknown. Does the system $$A(t)u'(t) + B(t)u + c(t)= 0$$ have a unique solution? Again, I don't know that $A(t)^{-1}$ is in $L^\infty.$

• Answer:

  Let $n=1$. Let $A(t)=1$, $B(t)=0$ and let $c(t)=0$ for all $t\in(0,T)$ for any $T>0$. Then $A(t)$ has an inverse for $t\in (0,T)$. Observe that $u(t)=k$ for all $t\in (0,T)$, for any $k\in \mathbb{R}$ solves the differential equation $$ A(t)u'(t) + B(t) + c(t) = 1*0 + 0 + 0 = 0 \ \ \forall t\in (0,T). $$ You may wish to impose boundary conditions on the problem if you seek a uniqueness result. I also recommend the first two chapters of the book by Coddington and Levinson "Theory of Ordinary DIfferential Equations" to address this type of question.