I am trying to solve an ode that uses some extremely large numbers and some extremely small numbers, namely

$$ e = 1.6\times 10^{-19}\\ E = 10^6\\ \tau = 6\times 10^{-24}\\ m = 9.1\times 10^{-31}\\ c = 3\times10^8\\ $$

I introduced some dimensionless constants, one in particular being $$ \varepsilon = \frac{\tau e}{cm} E \approx 4\times10^{-15} $$

MATLAB's ode45, ode15s, and ode23s cannot handle these numbers. Am I allowed to just set $\varepsilon = 10$? That number produces a pleasant graph with a nice solution, but I am not sure how to go from an $\varepsilon = 10$ world back to the original world.