System of differential equations with integral condition I am trying to solve differential equations of the following type
$$
f'(y)= A(y)f(y),
$$
where $y \in D$, $f: D \subseteq \mathbb{R} \to \mathbb{R}^n$, $A: D \to  \mathbb{R}^{n \times n}$ subject to $\int_{D} f_j(y) dy =1$, $j=1,...,n$. Is there any theory known for solving equations of this type, i.e. any existence and uniqueness theory?
 A: To answer this, it is useful to know that 
(1) there will be a vector-valued parameter for the general solution 
(2) the integral over $D$ of $f$ will depend continuously on these parameters, and 
(3) (a) it is easy to see that the $f=0$ is a solution, which also has 0 for the integral, while (b) there will be solutions where this integral will be arbitrarily large.  So existence should be guaranteed.  
Uniqueness on the other hand, maybe.  You have $n$ parameters, and $n$ constraints, so uniqueness is possible.  However, the constraint equations, while continuous on the parameters, are nonlinear in general, so there will not be a general theory to determine uniqueness.
A: You got a homogeneous linear system. Its solution can be written via a fundamental matrix as 
$$
f(y)=\Phi(y)f(a).
$$
Your condition then translates as
$$
{\bf 1}=\int_a^b\Phi(y)dy·f(a)
$$
and you get the usual conditions on solvability. Since $\Phi(a)=I$ the system matrix is invertible if $b$ is close enough to $a$, for general intervals, anything could happen.
