  The bouquet of circles $B_n$ and dipole graph $D_n$ are two important classes
of graphs in topological graph theory. For $n\geq 1$, we give an explicit
formula for the average genus $\gamma_{avg}(B_n)$ of $B_n$. By this expression,
one easily sees $\gamma_{avg}(B_n)=\frac{n-\ln n-c+1-\ln 2}{2}+o(1)$, where $c$
is the Euler constant. Similar results are obtained for $D_n$. Our method is
new and deeply depends on the knowledge in ordinary differential equations.
