I think Mathematica should have been to solve it.
Here is what I did: Solved this by hand using state space type approach. i.e. make 2 first order ODE's, from the second order ODE. These two first order ODE's can be solved by Mathematica on their own, using the initial conditions you gave above.
The issue is that, for the second ODE, it needed a specific set of assumptions to make it integrate it, and I think that is why DSolve could not do it. It needed these assumptions.
But taking these assumptions, after they are found, and passing them to DSolve, in the hope now DSolve can now solve the second order ODE, did not do the trick. This means, DSolve must be using some other method to solve this second order ODE, or it is not using the assumptions. Maple was able to solve the second order ODE as is with no assumptions, and it matches the solution using the state space method. Here are the steps done:
\begin{align*}
mx^{\prime\prime}\left( t\right) +ae^{bx^{\prime}\left( t\right) } &
=0\\
x\left( 0\right) & =0\\
x^{\prime}\left( 0\right) & =v_{0}%
\end{align*}
Let
\begin{align*}
x_{1}\left( t\right) & =x\left( t\right) \\
x_{2}\left( t\right) & =x^{\prime}\left( t\right)
\end{align*}
The initial conditions in new state variables are $x_{1}\left( 0\right)
=0$ and $x_{2}\left( 0\right) =v_{0}$. Taking derivatives of the above ODE's
give
\begin{align}
x_{1}^{\prime}\left( t\right) & =x_{2}\left( t\right) \tag{1}\\
x_{2}^{\prime}\left( t\right) & =-\frac{a}{m}e^{bx_{2}\left( t\right)
}\tag{2}%
\end{align}
The solution to the first ODE is
$$
x_{1}\left( t\right) =c_{0}+\int_{0}^{t}x_{2}\left( \tau\right) d\tau
$$
When $t=0,x_{1}\left( 0\right) =0$, hence $c_{0}=0$ and the above becomes
\begin{equation}
x_{1}\left( t\right) =\int_{0}^{t}x_{2}\left( \tau\right) d\tau\tag{3}
\end{equation}
The solution to the second ODE is, using $x_{2}\left( 0\right) =v_{0}$ is
DSolve[{x2'[t] == -a/m Exp[b x2[t]], x2[0] == v0}, x2[t], t]
$$
x_{2}\left( t\right) =-\frac{1}{b}\ln\left( \frac{bat}{m}+e^{-bv_{0}
}\right)
$$
Using the above, now the solution in Eq (3) can be found by integrating
$x_{2}\left( t\right) $
$$
x_{1}\left( t\right) =-\frac{1}{b}\int_{0}^{t}\ln\left( \frac{ba\tau}
{m}+e^{-bv_{0}}\right) d\tau
$$
The above can be integrated by Mathematica giving
$$
x_{1}\left( t\right) =-\frac{e^{-bv_{0}}}{ab^{2}}\left( \left(
abte^{bv_{0}}+m\right) \log\left( \frac{abt}{m}+e^{-bv_{0}}\right)
-abte^{bv_{0}}+bmv_{0}\right)
$$
But it needed assumptions to integrate the above. Here are the assumption (do not ask me how I found these, long story, trial and error and guessing :) to make the integral converges
Assuming[Element[{b, v0, m, a, t}, Reals] && a != 0 && b != 0 && m != 0 && t != 0 &&
m/(a b t) >= 0 && t > 0 && m > 0 && ((b < 0 && a < 0) || (b > 0 && a > 0)),
-1/b Integrate[Log[(b a tao)/m + Exp[-b v0]], {tao, 0, t}]]
So the two solution are found. They are again:
\begin{align*}
x_{1}\left( t\right) & =-\frac{e^{-bv_{0}}}{ab^{2}}\left( \left(
abte^{bv_{0}}+m\right) \log\left( \frac{abt}{m}+e^{-bv_{0}}\right)
-abte^{bv_{0}}+bmv_{0}\right) \\
x_{2}\left( t\right) & =-\frac{1}{b}\ln\left( \frac{bat}{m}+e^{-bv_{0}%
}\right)
\end{align*}
To verify with Maple:
Now, taking the assumptions used above to integrate for $x2(t)$ and pass them to DSolve:
$Assumptions = Element[{b, v0, m, a, t}, Reals] && a != 0 && b != 0
&& m != 0 && m/(a b t) >= 0 &&
m > 0 && ((b < 0 && a < 0) || (b > 0 && a > 0));
diffeq = m x''[t] == -a E^(b*x'[t]);
sol = DSolve[{diffeq, x[0] == 0, x'[0] == v0}, x[t], t]
(* {} *)
I also tried Assuming[....
Bottom line: I think DSolve should have been able to do this. Might require better assumption that the ones used above.
