This is similar to your previous code, this time with the additional constant (that does not seem to affect the result). There is not much to describe, really. See the documentation on Anonymous Functions and ode45 for those details.
mdot = @(t,m) -0.31458*sqrt(2E+5.*((1+(0.4-m)./0.5).^-1.4 - 1));
[t,m] = ode45(mdot, [0 0.5], 0.4);
plot(t, m, '-*b')
The ode45 call has as its arguments, the function to integrate ( mdot here ), the time span ( [0 0.5] here ), and the initial conditions ( 0.4 here, and only one because there is only one variable to integrate ). The figure plots m as a function of t plotting m(t) with a blue asterisk and drawing a blue line between them. The grid call adds grid lines to the plot.