One may simply use the theorem of Boltzmann's that is in [the very Wikipedia article you point to](http://en.wikipedia.org/wiki/Maximum_entropy_probability_distribution#Continuous_version).

Note that specifying the mean and variance is equivalent to specifying the first two raw moments - each determines the other (it's not actually necessary to invoke this, since we may apply the theorem directly to the mean and variance, it's just a little simpler this way).

The theorem then establishes that the density must be of the form:

$$f(x)=c \exp\left(\lambda_1 x + \lambda_2 x^2 \right)\quad \mbox{ for all } x > 0$$

Integrability over the positive real line will restrict $\lambda_2$ to be $\leq 0$, and places some restrictions on the relationships between the $\lambda$s (which will presumably be satisfied automatically when starting from the specified mean and variance rather than the raw moments).

To my surprise (since I wouldn't have expected it when I started this answer), this appears to leave us with a truncated normal distribution.

As it happens, I don't think I've used this theorem before, so criticisms or helpful suggestions on anything I haven't considered or have left out would be welcome.