  Consider n non-intersecting Brownian motions on $\mathbb{R}$, depending on
time $t \in [0,1]$, with $m_i$ particles forced to leave from $a_i$ at time
$t=0$, $1\leq i\leq q$, and $n_j$ particles forced to end up at $b_j$ at time
$t=1$, $1\leq j\leq p$. For arbitrary $p$ and $q$, it is not known if the
distribution of the positions of the non-intersecting Brownian particles at a
given time $0<t<1$, is the same as the joint distribution of the eigenvalues of
a matrix ensemble. This paper proves the existence, for general $p$ and $q$, of
a partial differential equation (PDE) satisfied by the log of the probability
to find all the particles in a disjoint union of intervals
$E=\cup_{i=1}^{r}[c_{2i-1},c_{2i}]\subset\mathbb{R}$ at a given time $0<t<1$.
The variables are the coordinates of the starting and ending points of the
particles, and the boundary points of the set $E$. The proof of the existence
of such a PDE, using Virasoro constraints and the multicomponent KP hierarchy,
is based on the method of elimination of the unwanted partials; that this is
possible is a miracle. Unfortunately we were unable to find its explicit
expression. The case $p=q=2$ will be discussed in the last section.
